# R4 Summary Integral Calculus

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Fundamental Concepts and Techniques of Calculus

R 4: Summary: Integral Calculus
Last updated: 041008

1. T HE R IEMANN I NTEGRAL :                                              (d) We say that a function f bounded on [a, b] is Rie-
mann integrable or simply integrable if I = I.
(a) Supremum, Inﬁmum: If A ⊆ R is a subset                                In this case we deﬁne
of the set of real numbers R, then we deﬁne
sup(A) to be the least upper bound of A if A is                                                    b
bounded above and ∞ it it is not bounded above.                                                        f (x) dx := I = I,
a
Similarly, we deﬁne inf(A) to be the greatest
lower bound of A if A is bounded below and                             called the deﬁnite(Riemann) integral or simply
−∞ if it is not bounded below.                                         the integral of f from a to b. Moreover, we de-
(b) Partition: Let [a, b] be a closed interval. A                           ﬁne
subset P := {x0 , x1 , . . . , xn } of [a, b] is called                                 a
a partition of [a, b] if                                                                    f (x) dx := 0
a
a = x0 < x1 < · · · < xn = b.                                                 a                                b
f (x) dx := −                    f (x) dx.
b                                a
(c) Upper Sums, Lower Sums:
i. Suppose the function f is bounded on [a, b]                  (e) A bounded function f on [a, b] is Riemann inte-
and P := {x0 , x1 , . . . , xn } is a partition of               grable if and only if for all ε > 0 there exists a
[a, b]. Then we set                                              partition P of [a, b] such that
Mi (f ) := sup{f (x) | x ∈ [xi−1 , xi ]}
U (f, P ) − L(f, P ) < ε.
mi (f ) := inf{f (x) | x ∈ [xi−1 , xi ]},

and ∆xi := xi − xi−1 , and call the sums                      (f) If f is (piecewise) monotonic (either increasing
or decreasing) on [a, b] then f is Riemann inte-
n                                         grable.
U (f, P ) :=         Mi (f )∆xi
i=1                                  (g) If f is (piecewise) continuous on [a, b] then f is
n                                       Riemann integrable.
L(f, P ) :=          mi (f )∆xi
(h) Suppose f is Riemann integrable on [a, b]. Then
i=1
for any ε > 0 there exists a partition P =
upper sum and lower sum of f with respect                       {x0 , . . . , xn } of [a, b] such that for any choice
to P , respectively.                                            of ξi ∈ [xi−1 , xi ]
ii. If P and Q are partitions and if P ⊆ Q
b                         n
then U (f, Q) ≤ U (f, P ) and L(f, Q) ≥
L(f, P ).                                                                        f (x) dx −                 f (ξi )∆xi < ε.
a                            i=1
iii. If P and Q are partitions, then L(f, P ) ≤
U (f, Q).                                                                            n
The sum i=1 f (ξi )∆xi is called a Riemann
iv. Let                                                               sum of f with respect to the partition P .

I := inf{U (f, Q) | Q partition of [a, b]}              2. E STIMATING D EFINITE I NTEGRALS :
I := sup{L(f, Q) | Q partition of [a, b]},                  Given an Riemann integrable function f deﬁned on
[a, b], choose n ∈ N and subdivide [a, b] in n subinter-
then I ≤ I.                                                 vals of equal length ∆x := b−a . Let xi := a + i ∆x.
n

1
(a) Left-Hand and Right-Hand Sums:                                       3. P ROPERTIES OF THE D EFINITE I NTEGRAL :
n−1                                     (a) Linearity: Suppose f and g are integrable and
LHSf (n) :=          f (xi )∆x                             α ∈ R, then
i=0                                                            b                                           b
n                                            i.                   αf (x) dx = α                               f (x) dx
RHSf (n) :=          f (xi )∆x.                                           a                                           a
B
i=1
ii.                    f (x) + g(x) dx
If the function f is differentiable on [a, b] and                                           a
b                                            b
|f (x)| ≤ B for all x ∈ [a, b] and some B ∈ R,                                                          =               f (x) dx +                                   g(x) dx
then                                                                                                            a                                            a
(b) Interval Property: Suppose f is integrable on
b
B(b − a)2                              the interval I and a, b, c ∈ I, then
f (x) dx − LHSf (n) ≤
a                                2n
c                               b                                            c
b
B(b − a)    2                                   f (x) dx =                      f (x) dx +                                   f (x) dx.
f (x) dx − RHSf (n) ≤                .                              a                               a                                            b
a                                         2n
(c) Order Properties:
(b) Trapezoidal Estimate:                                                          i. If the function f is integrable on [a, b]
1                                                       and f (x) ≥ 0 for all x ∈ [a, b], then
TRAPf (n) :=              LHSf (n) + RHSf (n) .                                   b
f (x) dx ≥ 0.
2                                                        a
ii. If the functions f and g are integrable on
If the function f is twice differentiable on [a, b]
[a, b] and f (x) ≤ g(x) for all x ∈ [a, b],
and |f (x)| ≤ B for all x ∈ [a, b] for some                                               b             b
b ∈ R, then                                                                       then a f (x) dx ≤ a g(x) dx.
iii. If f is integrable on [a, b], then |f | is inte-
b
B(b − a)3                           grable on [a, b] and
f (x) dx − TRAPf (n) ≤                     .
a                                           12n2                                                       b                                         b
f (x) dx ≤                                |f (x)| dx.
(c) Midpoint Sum:                                                                                          a                                         a
Let µi := 1 (xi−1 + xi ) = a + (i − 1 )∆x de-
2                           2
note the midpoint of the ith subinterval of [a, b].                           iv. If f is integrable on [a, b] and m the min-
Then                                                                              imum and M the maximum value of f on
n
[a, b] then
MPSf (n) :=          f (µi )∆x.                                                                               b
i=1                                                        m(b − a) ≤                              f (x) dx ≤ M (b − a).
a
If the function is twice differentiable on [a, b]
and |f (x) ≤ B for all x ∈ [a, b] for some                               (d) Symmetry: Suppose f is integrable on the
b ∈ R. then                                                                  symmetric interval [−a, a]. Then,

b
i. if f is odd on [−a, a], then
B(b − a)3
f (x) dx − MPSf (n) ≤           .                                                                       a
a                                  24n2
f (x) dx = 0;
−a
(d) Simpson’s Rule:
1“                                ”                     ii. if f is even on [−a, a], then
SIMPf (2n) : =           2 · MPSf (n) + TRAPf (n)
3                                                                                 a                                                a
n
X“                                ” ∆x                                                   f (x) dx = 2                                  f (x) dx.
=      f (xi−1 + 4f (µi ) + f (xi )      .                                         −a                                         0
i=1
6

If the function f is four times differentiable on                       (e) Average Value of a Function: If f is integrable
[a, b] and |f (4) (x)| ≤ B for all x ∈ [a, b] for                            on [a, b], then
some B ∈ R, then                                                                                                                    b
1
b
f (x) dx
B(b − a)5                                                               b−a              a
f (x) dx − SIMPf (n) ≤           .
a                                  180n4
is the average value of f on the interval [a, b].

2

(f) The Mean-value Theorem for Integrals: If f is                                                       g(x) r+1
+ C, r = −1

r
continuous on [a, b], then there exists a number                            (b)    g(x) g (x) dx =
ξ ∈ [a, b] such that                                                                                r+1
ln |g(x)| + C,   r = −1
b
1                                             (c)    cos x dx = sin x + C
f (ξ) =                          f (x) dx.
b−a           a
sin x dx = − cos x + C
(g) Second Mean-value Theorem for Integrals:
If f and g are continuous on [a, b] and g(x) ≥ 0                             (d)    sec x dx = ln | sec x + tan x| + C
for all x ∈ [a, b], then there exists a number
ξ ∈ [a, b] such that                                                                csc x dx = − ln | csc x + cot x| + C.
b                                           b
f (x)g(x) dx = f (ξ)                        g(x) dx.       (e)    sec2 dx = tan x + C
a                                           a

csc2 x dx = − cot x + C
(h) Anti-derivative:     A function F is an anti-
derivative or primitive or indeﬁnite integral of
the function f on an interval I if F (x) = f (x)                             (f)    sec3 x dx =   1
2   sec x tan x +   1
2   ln | sec x + tan x| + C
for all x ∈ I.
csc3 x dx = − 1 csc x cot x −
2
1
2   ln | csc x + cot x| + C
4. T HE F UNDAMENTAL T HEOREM OF C ALCULUS :
(g)    sec x tan x dx = sec x + C
(a) If f is continuous on [a, b] then the function
x                                        csc x cot x dx = − csc x+C
F (x) :=                 f (t) dt
a
(h)    ex dx = ex + C
is differentiable on (a, b) and                                                                ax
ax dx =        +C
x                                                       ln a
d
F (x) =                             f (t) dt            = f (x)                  1
dx        a                                           (i)          dx = tan−1 x + C
1 + x2
for all x ∈ (a, b).                                                                     1
(j)   √        dx = sin−1 x + C
(b) If the function f is continuous on [a, b] and F                                       1 − x2
any anti-derivative of f on [a, b], then                                                 1
(k)     √        dx = sec−1 x + C
x x2 − 1
b                                                          b
f (x) dx = F (b) − F (a) = F (x) .                              (l)   cosh x dx = sinh x + C
a                                                              a
sinh x dx = cosh x + C
(c) If f is continuous on [a, b] and ψ1 and ψ2 are
differentiable on (a, b) then
(m)    sech2 x dx = tanh x + C
ψ2 (x)
G(x) :=                         f (t) dt                        csch2 x dx = − coth x + C
ψ1 (x)

ist differentiable on (a, b) and                                             (n)    sech x tanh x dx = − sech x + C

d         ψ2 (x)                                                csch x coth x dx = − csch x + C
G (x) =                                 f (t) dt
dx        ψ1 (x)
(o)    cosn x dx =   1
n   cosn−1 x sin x +        n−1
n      cosn−2 x dx
= f ψ2 (x) ψ2 (x) − f ψ1 (x) ψ1 (x).
sinn x dx = − n sinn−1 x cos x +
1                             n−1
n      sinn−2 x dx
5. BASIC I NTEGRATION F ORMULAS :
 r+1
x                                                                (p)    tann x dx =    1
n−1    tann−1 x −            tann−1 x dx
+ C, r = −1
(a)    xr dx = r + 1

ln |x| + C, r = −1                                                      cotn x dx = − n−1 cotn−1 x −
1
cotn−1 x dx

3
(q)     secn x dx =        1
secn−2 x tan x +              n−2
secn−2 x dx   (S TEP 2:) Decompose p(x)/q(x) by replac-
n−1                                 n−1
ing each linear factor of q which has the form
k
cscn x dx = − n−1 cscn−2 x cot x +
1                                        n−2
n−1    csc n−2
x dx (ax + b) (with multiplicity k ≥ 1) by
A1       A2                Ak
6. I NTEGRATION RULES :                                                                          +          + ··· +
ax + b (ax + b)2         (ax + b)k
(a) Linearity:                                                                      and each quadratic factor of q which has the
form (ax2 + bx + c)k (with multiplicity k ≥ 1)
i.       αf (x) dx = α           f (x) dx                                    by

ii.           f (x) + g(x) dx =                                                    A1 x + B1              Ak x + Bk
+ ··· +
ax2 + bx + c         (ax2 + bx + c)k
=      f (x) dx +             g(x) dx.
(S TEP 3:) Compute the undetermined coefﬁ-
(b) Substitution Rule:                                                              cients Aj , Bj in the numerators using “compar-
b                                   g(b)                            ing coefﬁcients”, limit techniuques etc.
i.           f g(x) g (x) dx =                      f (y) dy                 (S TEP 4:) Solve the resulting integrals which
a                                g(a)                                   are of the form
ii.       f g(x) g (x) dx =               f (y) dy                                     Aj                                Aj x + bj
dx,                                   dx
iii. Trigonometric Substitutions:                                                   (ax + b)j )                       (ax2 + bx + c)j
√
√ If the integrand involves
√                  a2 − x2 ,
7. A PPLICATIONS :
x2 − a2 or a2 + x2 , substitute x =
a sin θ, x = a sec θ, x = a tan θ, respec-                           (a) Area between Two Curves:            If f and g are
tively.                                                                  continuous functions on [a, b] such that g(x) ≤
iv. Rationalizing Substitutions:                                             f (x) for all x ∈ [a, b], then the region {(x, y) |
If the integrand involves an expression of                             a ≤ x ≤ b, g(x) ≤ y ≤ f (x)} is called the re-
the form n f (x) substitute y = n f (x) or                               gion between f and g over [a, b]. Then the area
equivalently, y n = f (x).                                               differential at x ∈ [a, b] is given by
If the integrand is a rational expression in
sin x and cos x, substitute y = tan(x/2),                                          dA(x) = f (x) − g(x) dx
which implies                                                            and the total area by
1 − y2                             2y                                     b                  b
cos x =       ,         sin x =                                      A=              dA =               f (x) − g(x) dx.
1 + y2                           1 + y2
a                  a
2 dy
dx =
1 + y2                                               (b) Volume by Slicing:
i. Cavalieri’s Principle: Given a solid Ω
(c) Integration by Parts:                                                                with continuous cross-sections A(x) per-
b
pendicular to a coordinate axis whose
i.           f (x)g(x) dx =                                                       points are labeled by x. Then the volume
a
b           b                                        differential of Ω at the point x is given by
= f (x)g(x)       −           f (x)g (x) dx
a       a                                                              dV (x) = A(x) dx
ii.       f (x)g(x) dx =                                                           and if a < b are points on the axis, then the
volume of Ω between a and b is given by
= f (x)g(x) −         f (x)g (x) dx
b                  b
(d) Integration by Partial Fractions: If the inte-                                               V =                dA =               A(x) dx.
grand is a proper rational function of the form                                                         a                  a

p(x)/q(x) where p and q are polynomials (if it                                   ii. Disc/Washer Method: Let c ∈ R and f
is not proper, ﬁrst use long division) follow this                                   and g continuous on [a, b] such that c ≤
procedure:                                                                           g(x) ≤ f (x) for all x ∈ [a, b]. Moreover,
(S TEP 1:) Factor q into a product of linear and                                     let Ω denote the solid generated by revolv-
irreducible (over R) quadratic factors.                                              ing the region between f and g over [a, b]

4
about the axis y = c. Then the volume dif-                            iii. If f is continuously differentiable, then
ferential of Ω at the point x ∈ [a, b] is given
by                                                                                                                               2
ds(x) :=                1 + f (x)             dx
2                2
dV (x) = π f (x) − c                    − g(x) − c       dx
is the arc length differential of the curve Cf
and the total volume of Ω by                                               at x. The total length of the curve Cf is
given by
b
V =             dV                                                                          b                  b
a                                                                                                                             2
b
s=               ds =               1 + f (x)              dx.
2                2                                  a                  a
=π                   f (x) − c       − g(x) − c       dx
a
(e) Area of a Surface of Revolution: Let c ∈ R
(c) Volume by Shells: Let c ∈ R, c ≤ a < b,                                   and let f be continuously differentiable on [a, b]
and let f and g be continuous on [a, b] such that                         and f (x) ≥ c for all x ∈ [a, b]. Moreover, let
g(x) ≤ f (x) for all x ∈ [a, b]. Moreover, let Ω                          Γ denote the surface generated by revolving the
denote the solid generated by revolving the re-                           curve Cf about the axis y = c. Then the surface
gion between f and g over [a, b] about the axis                           area differential of Γ at x ∈ [a, b] is given by
x = d. Then the volume differential of Ω at
x ∈ [a, b] is given by                                                        dS = 2π f (x) − c ds(x)
dV (x) = 2π(x − d) f (x) − g(x) dx,                                            = 2π f (x) − c                      1 + f (x)
2
dx
and the total volume by
and the total surface area by
b
V =            dV                                                                 b
a
S=             dσ
b
a
= 2π                (x − d) f (x) − g(x) dx,                                   b
a                                                                                                                     2
=           2π f (x) − c                1 + f (x)              dx.
a
(d) Arc Length:
i. Arc, Smooth Curve: If f is continuously                          (f) Mass from Density:
differentiable on [a, b] (i.e. f is continuous)
then the curve                                                         i. Mass of a a Rod: Suppose the projection
of the material solid Ω onto the real num-
C := Cf := { x, f (x) | a ≤ x ≤ b}                                      ber line is given by the interval [a, b], and
suppose δ(x) is the density of the solid at
is called an arc or a smooth curve.
ii. Arc Length: Let Cf be an arc, deﬁned                                      x ∈ [a, b] (units: mass per unit length).
by the function f on [a, b], and let P :=                                 Then the mass differential of Ω at x ∈ [a, b]
{x0 , . . . , sn } be a partition of [a, b]. More-                        is given by
over, let
q               `                    ´2                                                 dm(x) = δ(x) dx,
∆sj : = (xj − xj−1 )2 + f (xj ) − f (xj−1 )
q    `      ´2
= 1 + f (ξi ) ∆xj                                                   and the total mass of Ω by
for some ξj ∈ (xj−1 , xj ). Then,                                                                    b                  b
n                                                  m=                 dm =               δ(x) dx.
a                  a
s(P ) :=         ∆sj
j=1
ii. The Mass of a Plate: Suppose the projec-
gives the length of the polygonial arc in-                                 tion of the material solid Ω in the plane is
scribed in Cf relative to the partition P .                                the region between f and g over the interval
If the set {s(P ) | P partition of [a, b]} is                              [a, b]. Moreover, suppose that the density
bounded above, then Cf is called rectiﬁable                                of the solid is given by the (1-dimensional)
and we deﬁne the length s of Cf by                                         function δ : [a, b] → R (units: mass per
unit area); note: we assume the density is
s := sup{s(P ) | P partition of [a, b]}.                                constant along the lines x = c for c ∈ [a, b].

5
Then the mass differential of Ω at x ∈ [a, b]                and the total moment of Ω by
is given by
b

dm(x) = f (x) − g(x) δ(x) dx,                               M=                      dM
a
b                             b
and the total mass of Ω by                                           =                   x dm(x) =                     xδ(x) dx.
a                             a
b
m=             dm                                       The center of mass of Ω is the unique point
a
b
x ∈ R such that xm = M where m denotes
=            f (x) − g(x) δ(x) dx.                   the total mass of Ω. Thus
a
b
iii. The Mass of a Wire: Suppose the axis of                                                 dM                   xδ(x) dx
a
the wire W is a smooth curve Cf deﬁned by                           x=                       =            b
.
the function f continuously differentiable                                              dm                    δ(x) dx
on [a, b]. Moreover, suppose that δ(x) is                                                             a
the density of W at x ∈ [a, b] (units: mass
ii. The Centroid of a Rod: If the density of
per unit length). Then the mass differential
the rod Ω is constant, it is called a uniform
of W at x ∈ [a, b] is given by
or homogeneous rod. Its center of mass x is
dm(x) = δ(x) ds(x)                                       called the centroid of the rod. Clearly,

2                                                  1
= δ(x) 1 + f (x)                   dx,                                       x=     (a + b).
2
and the total mass of W by
iii. The Center of Mass of a Plate: Suppose
b                                             the Plate Ω is given by the continuous func-
m=             dm                                        tions f and g on [a, b] and the density func-
a                                                 tion δ on [a, b]. We consider the moment
b
Mx about the x-axis and the moment My
=           δ(x) ds
a                                                 about the y-axis. Then the moment differ-
b
2
entials of Ω at x with respect to the x-axis
=           δ(x) 1 + f (x)             dx.            and y-axis are given by
a

iv. Three Dimensions: Suppose Ω is a mate-                                    ˆ
dMx (x) = y dm(x)
rial solid desribed by the 3-dimensional re-                             = 1 f (x) + g(x) dm(x),
2
gion B in R3 with density function δ as-                                   ˆ
dMy (x) = x dm(x) = x dm(x),
signing to each (x, y, z) ∈ B its density
δ(x, y, z). Then the mass differential of Ω                                       ˆ      ˆ
respectively, where x and y denote the x-
at (x, y, z) is given by                                     and y-coordinate of the center of mass of
the (uniform rod) mass differential dm(x)
dm(x, y, z) = δ(x, y, z) dV (x, y, z)
at x. The total moments of Ω with respect
and the the total mass of Ω by                               to the x- and y-axis are given by
b
m=             dm =          δ(x, y, z) dV.             Mx =                dMx
B                 B                                          a
b
1                            2                      2
(g) Center of Mass and Centroid:                                         =   2                    f (x)       − g(x)                 δ(x) dx,
a
i. The Center of Mass of a Rod: Given a rod                                     b
Ω with density function δ : [a, b] → R. We                   My =                dMy
consider the moment of Ω with respect to                                 a
b
the point x = 0. The moment differential
=               x f (x) − g(x) δ(x) dx,
of Ω at x ∈ [a, b] (the moment of the x-th                               a
slice of Ω) is given by
respectively. The center of mass of Ω is
dM (x) = x dm(x) = xδ(x) dx                             the unique point (x, y) ∈ R2 such that

6
xm = My and ym = Mx , where m de-                                             vii. Center of Mass in Three Dimensions:
notes the total mass of Ω. Thus                                                    Suppose Ω is a material solid desribed by
the 3-dimensional region B in R3 with den-
dMy               ˆ
x dm                                    sity function δ assigning to each (x, y, z) ∈
x=                    =                        ,                          B its density δ(x, y, z). Let (ˆ, y , z ) ∈ R3
x ˆ ˆ
dm                dm                                      denote the center of mass of the mass differ-
ential dm at (x, y, z) ∈ B, then the center
dMx               y dm
ˆ                                       of mass (x, y, z) of Ω is given by
y=                   =                        ,
dm                dm                                                    ˆ
x dm           ˆ
x δ(x, y, z) dV
B              B
x=                =
dm              δ(x, y, z) dV
B
respectively.
iv. The Centroid of a Region: If the density                                                        ˆ
y dm           ˆ
y δ(x, y, z) dV
B              B
of the plate Ω is constant, we call its center                                       y=                =
of mass, the centroid of the region covered                                                     dm              δ(x, y, z) dV
B
by the plate.
v. The Center of Mass of a Wire: Suppose                                                           ˆ
z dm           ˆ
z δ(x, y, z) dV
B              B
the axis of the wire W is the smooth curve                                           z=                =
Cf , f continuously differentiable on [a, b]                                                    dm              δ(x, y, z) dV
with density δ on [a, b]. The mass differ-                                                                 B

x ˆ
ential dm(x) has center of mass (ˆ, y ) =                                        Let Γ denote the surface of B and let again
(x, y). Then the moment differential of W                                         x ˆ ˆ
(ˆ, y , z ) denote the center of mass of the
at x with respect to the x-axis and y-axis is                                    mass differential dm = δ(z, y, z) dS at
given by                                                                         (x, y, z) ∈ Γ. Then the center of mass
(x, y, z) of Γ is given by
ˆ
dMx (x) = y dm(x)
= y dm(x) = y δ(x) ds,
ˆ
x dm           ˆ
x δ(x, y, z) dS
ˆ
dMy (x) = x dm(x)                                                               x=     S
=   S

= x dm(x) = x δ(x) ds,                                                             dm              δ(x, y, z) dS
S
respectively, and the total moment of W
with respect to the x-axis and y-axis by                                                         ˆ
y dm           ˆ
y δ(x, y, z) dS
S              S
y=                =
b                 b
dm              δ(x, y, z) dS
Mx =             dMx =             y δ(x) ds(x).                                                         S
a                 a
b              b                                                                ˆ
z dm           ˆ
z δ(x, y, z) dS
My =             dMy =             x δ(x) ds(x),                                  z=     S
=   S
a                 a
dm              δ(x, y, z) dS
respectively. The center of mass (x, y) of                                                                  S
W is therefore given by                                                      viii. Symmetry and Centroid:
b                                  If the (1-, 2-, 3-dimensional) region Ω has
dMy               x dm
ˆ                        x δ(x) ds                      axis of symmetry then the centroid z of Ω
a
x=                   =                 =            b
,                lies on .
dm               dm                           δ(x) ds           (h) Fluid Pressure:
a
b                                  i. The pressure p exerted by a ﬂuid with
dMx               y dm
ˆ                        y δ(x) ds                      weight-density ω at depth h is given by
a
y=                  =                 =         b
,
dm               dm                                                                           p = ωh.
δ(x) ds
a
ii. The force F exerted by a ﬂuid on a (hori-
vi. The Centroid of an Arc: If the density of                                         zontal) surface of area A at depth h is
a wire is constant, it is called the centroid
of the arc covered by the wire.                                                                 F = pA = ωhA.

7
(i) Work:                                                                   and the future value of the entire money
i. The work W performed by an object moved                             stream P (t) at time T is
a distance d along a straight line by a con-
T                   T
stant force F is
B=             dB(t) =             P (t)er(T −t) dt.
W = F d.                                                  0                   0

(j) Money Stream:                                           8. T HE T HEOREMS OF PAPPUS :
i. A money stream is a continuous function
representing the ﬂow of money as a func-               (a) The First Theorem of Pappus: Suppose the
tion of time (units: dollars per unit time).               solid Ω is generated by revolving the region R
ii. Future Value:                                              about the line that does not intersect R. Let r
A. Suppose an amount P is deposited into                  denote the distance of the centroid of R from .
an account at r percent interest com-                 Then
pounded continuously. Then its value
V = 2πrA,
B after T years is
B = P erT ,                              where A denotes the area of R.
called the future value of P after T               (b) The Second Theorem of Pappus: Suppose the
years at rate r.                                       surface Γ is generated by revolving the arc C
B. Now suppose a money stream P (t)                       about the line that does not intersect C. Let r
is deposited into an account at r                      denote the distance of the centroid of C from .
percent interest compounded continu-                   Then
ously. Then the “future value differen-                                        S = 2πrs,
tial” dB(t) at time t is
dB(t) = P (t) dt er(T −t)                      where s denotes the length of C.

8

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