Calculus Multiple Integration Homework Problems Section by sanmelody


									Calculus 3

                           14 Multiple Integration

Homework Problems
Section 14.1
   1, 6, 11, 16, 21, 27, 31, 37, 45, 53, 58, 59, 63, 67, 69, 73, 77, 79
Section 14.2
   3, 5, 9, 14, 19, 25, 30, 35, 40, 41, 49, 55, 58, 65, 68, 71, 74, 76, 80, 83
Section 14.3
   1 –8, 11, 17, 21, 25, 31, 39, 45, 49, 51, 53, 58
Section 14.4
   3, 7, 12, 17, 22, 29, 49, 54, 57
Section 14.5
   2, 7, 12, 17, 25, 27, 31, 35, 40
Section 14.6
   5, 9, 14, 21, 25, 30, 31, 35, 37, 41, 55, 57, 61, 63, 71
Section 14.7
   4, 11, 16, 19, 24, 29, 33, 35, 37, 42, 46
Section 14.8
   2, 7, 9, 13, 18, 22, 23, 25, 27, 30, 31

  Iterated integrals
  Integrals of two variables
  Riemann sums
  Su¢ cient condition for integrability
  Arithmetic properties
  Evaluating double integrals
  Integrating with polar coordinates
  Riemann sums for polar sectors
  Center of mass and moments of inertia
  Moment of inertia
  Surface area
  Triple integrals
  General change of coordinates
  Transformation of a rectangle
  Review: Simpson’ rule
  Review: Wallis’formula

    The primary focus of our e¤orts to this point in time has been to discuss
di¤erentiation, and in this chapter we will concern ourselves with integration.
To some extent, just like the notion of partial derivative, it is rather easy to
progress in a certain direction. The concept of iterated integral, which is the
standard approach to discussing integration involving more than one variable,
is the pure analogue to partial di¤erentiation.

Iterated integrals
   With an iterated integral, we hold all variables …xed but one for each part of
the iteration. For a function f (x; y) we can integrate, for example, by holding
y constant and integrating with respect to x:
                                         Z       b
                                                     f (x; y) dx                (1)
It is easy to give examples using polynomials.
    Because y is being treated as a constant, we can allow the limits of integration
a, and b to be functions of y:

                                                 a = a (y)                      (2)

                                                 b = b (y)                      (3)
Then we can integrate over y to obtain an iterated integral:
                              Z      d   Z   b(y)
                                                       f (x; y) dxdy            (4)
                                 c       a(y)

Similarly, we can have an iterated integral in which y is integrated …rst, with x
held …xed, and then we integrate over x:
                             Z     m     Z    h(x)
                                                       f (x; y) dydx            (5)
                               k             g(x)

   Together, the two integrations determine a region of integration in the xy-
plane. For example, the region associated with (4) is

                 c < y < d and, for a …xed y, a (y) < x < b (y)                 (6)
    Due to this, we can interpret the case when f (x; y) = 1 as giving the area
between two curves in the plane. Actually, sometimes we can reverse the order
of integration to …nd two di¤erent ways of representing the area.

Integrals of two variables
   Section 14.2 starts with the double integral in which we consider Riemann
sums. Just as one …nds the "area under a curve" for a function y = f (x)
of a single variable, we can describe what we mean by area under a surface

z = f (x; y) for a function of two variables. It is gratifying that the arithmetic
properties such as linearity for an integral of a function of one variable carry
over to a double integral, i.e. an integral of a function of two variables. Certain
volumes of solids, in which the surfaces involved are plane, are easy to evaluate.
The ideas that are involved in these simple computations lead to Fubini’ theo-
rem, Theorem 14.2, which gives the main connection between iterated integrals
and double integrals.

Riemann sums
   When we see the need to connect the di¤erent orders of iterated integration
over a region, then we see that it becomes important to de…ne the integral:
                                     f (x; y) dA                            (7)
Where R is some region in the xy-plane. The easiest, and most obvious way of
doing this is to …rst partition the region R into small rectangles

                                  P : R1 , R2 , ... Rn                          (8)
We can de…ne inner and outer partitions. The norm, j P j, of the partition is
the area of the largest rectangle in the partition.

                 A (Ri ) = area of rectangle Ri for i = 1; 2; :::n              (9)
and choose points in each rectangle:

               (x1 ; y1 ) in R1 , (x2 ; y2 ) in R2 , ... (xn ; yn ) in Rn      (10)
De…ne the value of the function at each of these points (these are the "heights"
of prisms de…ned by the rectangles, where we allow negative heights to indicate
that the prism is below the xy-plane):

               f1 = f (x1 ; y1 ) , f2 = f (x2 ; x2 ) , ... fn = f (xn ; yn )   (11)
    Having this setup, we can now de…ne a Riemann sum associated with the
integral (7):
                                            fi A (Ri )                         (12)

    For numerous types of functions f (x; y), we can de…ne a limiting process
for the Riemann sum, to unambiguously specify the value of a double integral.
This is notably the case for continuous functions, where the region R is a closed
and bounded set in the xy-plane. Then we are able to de…ne
                       Z                               n
                             f (x; y) dA =     lim             fi A (Ri )      (13)
                         R                   j P j!0

where we are only considering one class of partition, inner partitions or outer
partitions. It is then a goal to consider when the Riemann integral will be
consistent with either class.

    Clearly, if f (x; y) > 0, we can interpret the double integral (13) as a volume.
It is simple to construct examples using quadratic polynomials, i.e. computing
volumes whose bounding surfaces from above are part of a quadric surface. Let
y = f (x; y) be a non-negative function on the region R. Then we call
                                           f dA
the volume of the solid between the region R and the graph of f .

Su¢ cient condition for integrability
   If the integral (13) exists (and is …nite), we say that f is integrable on
R. If we restrict our attention to polynomials (or more generally continuous
functions), and assume R to be closed and bounded (so that the function values
are bounded, and the Riemann sums are …nite), then a su¢ cient condition is
not hard to …nd. We merely need to restrict ourselves to unions of a …nite
number of nonoverlapping subregions, each of which is vertically or horizontally
   If R is de…ned by a x b and gL (x) y gU (x), for a …xed value of x,
where gL and gU are continuous on [a; b], then the region R is vertically simple.
There is an analogous de…nition for a horizontally simple region.

Arithmetic properties
    The arithmetic properties of double integrals are associated with integrations
of polynomials, and if we wish to be a bit more general, of rational functions,
or continuous functions. For simplicity, assume the following functions are
continuous functions, and the region R over which we are integrating is a closed
and bounded region in the plane (and composed of nonoverlapping vertically
and horizontally simple regions.
    First, we note that double integrals preserve the vector structure of functions,
i.e. they are linear
                  Z                           Z            Z
                     [c1 f1 + c2 f2 ] dA = c1   f1 dA + c2   f2 dA              (14)
                   R                           R           R
where c1 and c2 are scalars and f1 and f2 are functions. There is also a positivity
                 jf j dA 0 and equality holds if and only if f = 0             (15)
If we have two nonoverlapping regions R1 and R2 that make the region R, then
(inherited from the de…nition of integration in terms of partitions):

                         Z              Z             Z
                               f dA =        f dA +        f dA               (16)
                           R            R1            R2

Due to the positivity property, we have the following ordering property:
                                     Z          Z
                        if f g then     f dA       gdA                        (17)
                                             R            R

Evaluating double integrals
    If we restrict ourselves to polynomials (more generally, continuous functions),
on horizontally or vertically simple regions that are closed and bounded, we
can use Fubini’ theorem to convert a double integral into an iterated integral.
Fubini’ theorem is Theorem 14.2 on page 994. Example 4 on page 996 is an
interesting example, because it shows that the choice of order of integration can
be important. Example 5 is also worth looking at, because it shows how one
can obtain a description for the region R in some situations. Note that this
last example applies Wallis’formula from page 536.

Integrating with polar coordinates
   An important consideration involved in integration of a function of one vari-
able is the change of variable formula
                         Z             Z
                            f (u) du = f (u (x)) dx                        (18)
which is obviously related to the chain rule. We do not consider the general
change of variable formula for double integrals in Section 14.3, but instead re-
strict ourselves to changing from rectangular to polar coordinates. Integrations
over polar coordinates can be split up into integrations over r-simple regions
and -simple regions.

Riemann sums for polar sectors
   A polar sector is a region

                               a < r < b and c <          <d                  (19)
where for simplicity we require [c; d] to be contained within [0; 2 ]. From this,
we can immediately de…ne an r-simple region, where varies between …xed c
and d, and for …xed , r varies between the two continuous functions rL ( )
and rU ( ). We can make an analogous de…nition for -simple regions. Then
we can consider the integration of continuous functions over nonoverlapping
r-simple and -simple regions. The area of a polar sector of the form (19)
is approximately r r , where r = b a and                = d c, and r is some
value selected between a and b, provided that the sector is small (which is a
satisfactoryR assumption for Riemann sums). This leads to Theorem 14.3 for
expressing R f (x; y) dA in polar coordinates. For example, if R is the polar
sector (19),

                Z                           Z    d   Z   b
                     f (x; y) dA =                           f (r cos ; r sin ) rdrd   (20)
                 R                           c       a
We can obviously write analogous expressions for r-simple of -simple regions.
    Example 3 on page 1004 is a nice example of the application of polar co- aRproblem that could not be practically integrated using x and y
coordinates. R dA is the area of region R. Example 4 provides a nice instance
of this applied to polar coordinates. Notice that
                       Z     d   Z   r( )                    Z     d
                                                                       1        2
                                            rdrd =                       (r ( )) d     (21)
                         c       0                             c       2

Center of mass and moment of inertia
    Section 14.4 is an application of double integrals to center of mass and mo-
ment of inertias of plane laminas. This allows the development of a physical
picture for double integration, and in many cases permits us to use notions
of symmetry to partially determine quantities such as centers of mass without
explicitly carrying out integral expressions.

    We de…ne mass on the basis of a mass density (x; y), which is a non-negative
integrable function, over a closed, bounded region R such as we have discussed
before (except now we will refer to it as a plane lamina). The mass m of the
lamina R is just the integral of over R.
    We de…ne the moments of mass Mx and My , as the integrals, respectively,
of x and y over R. The quantity

                                                               Mx My
                                 (xcm ; ycm ) =                  ;                     (22)
                                                               m m
is called the center of mass or centroid of the lamina. We call Mx and My …rst

Moment of inertia
    We de…ne second moments Ix and Iy to be the integrals over R, respectively,
of x2 and y 2 . The sum of Ix and Iy is called the polar moment of inertia and
is denoted I0 . Relative to this, we can de…ne a radius of gyration, rgyr :
                                   rgyr =                                   (23)
where I is the moment of inertia. When I = Ix , we denote rgyr as a y with a
double-bar above it, etc.

Surface area

    Section 14.5 concerns another application to the computation of surface area.
This is a simple application of the idea that if u and v are vectors along adjacent
sides of a parallelogram, then the area of the parallelogram is ju vj.
    Let y = f (x; y) and consider the points (x; y), (x + dx; y), and (x; y + dy)
(points in close proximity). The three points

(x; y; f (x; y)) , (x + dx; y; f (x; y) + fx (x; y) dx) , (x; y + dy; f (x; y) + fy (x; y) dy)
de…ne a parallelogram that approximates a region of the surface, with vectors

                       (dx; 0; fx (x; y) dx) , (0; dy; fy (x; y) dy)                 (25)
de…ning the sides of the parallelogram. Thus means that a section of surface
has the approximate area
                                2             3
                                   i   j   k
                           det 4 dx 0 fx dx 5                           (26)
                                   0 dy fy dy
Therefore, it is not surprising that we de…ne the surface area over the region R
to be
                           Z q
                                          2       2
                                 1 + (fx ) + (fy ) dxdy                      (27)
in cases where it is meaningful to do so.
    Example 5 on page 1021 is a useful example. It illustrates an application
of Simpson’ rule.

Triple integrals
    Of course, there is no reason, once we understand, to con…ne ourselves to
double integrals. It is a simple matter to extend certain elements of our discus-
sion to "triple integrals". This is done in Section 14.6. What are the analogies
that can readily be pursued? The idea of area computed by double integrals
can be extended to volume from triple integrals. The volume under a surface
can be extended to a corresponding concept for triple integrals. As this involves
four dimensions, the authors avoid discussion of this point. Finally, the center
of mass or moments of inertia of a plane lamina can be extended, using triple
integrals, to similar concepts for volumes.

Change of variables from rectangular to spherical or cylindrical coor-
    Another analogy is pursued in Section 14.7, where change of coordinates from
rectangular to spherical or cylindrical is discussed. This of course corresponds
to change of coordinates to polar coordinates for double integrals.Many integrals
can be more easily evaluated in cylindrical coordinates or spherical coordinates
than in rectangular coordinates. The cylindrical area element is

                                           drd dz                                     (28)
The spherical area element is

                                        sin d d d                                     (29)

   General change of coordinates
   In the …nal section, Section 14.8, the problem of general change of coordi-
nates is considered. The crucial quantity involved is something that follows
how a tiny square in the xy-coordinate system is transformed under a coor-
dinate transformation to some coordinates (u; v). Such a tiny square in the
xy-coordinate system will essentially become, under many conditions, including
polar coordinates (r; ) to a small parallelogram in the uv-plane. It is a determi-
nant that measures this shift. The determinant is called the Jacobian, and the
whole idea can be extended to any …nite number of variables. Polar, cylindrical
and spherical coordinates are by far the most important coordinate frames after
the rectangular. However, more general transformations sometimes need to be
considered, and this is particularly important for theoretical ideas.

    Transformation of a rectangle
    Consider the three vertices of a small parallelogram in uv-space, (u; v),
(u + dux ; v + dvx ) and (u + duy ; v + dvy ), corresponding to the rectangle ver-
tices (x; y), (x + dx; y) and (x; y + dy) in xy-space, i.e.
                                      @u              @v
                           dux =         dx and dvx =    dx                           (30)
                                      @x              @x
                                      @u              @v
                            duy =        dy and dvy =    dy                           (31)
                                      @y              @y
with area

            (@u=@x) dx (@v=@x) dx                        @u=@x       @v=@x
    det                                       = det                            dxdy   (32)
            (@u=@y) dy (@v=@y) dy                        @u=@y       @v=@y

   The determinant

                         @ (u; v)              @u=@x      @v=@x
                                  = det                                               (33)
                         @ (x; y)              @u=@y      @v=@y
is called the Jacobian, and we have from the above considerations, the general
change of variable formula

      Z                           Z
                                                                     @ (u; v)
                f (u; v) dudv =            f (u (x; y) ; v (x; y))            dxdy    (34)
       R(u;v)                     R(x;y)                             @ (x; y)

where R (x; y) is the region in the xy-plane that corresponds to the region
R (u; v) in the uv-plane.

    Review: Simpson’ rule s
    Calculators can perform numerical integrations. One practical algorithm
                   s                            s
for this is Simpson’ rule. The basis of Simpson’ rule is Theorem 4.17 on page
311, which is just the integral of a quadratic polynomial between two points
expressed in terms of the polynomial:

                                    p (x) = Ax2 + Bx + C                                   (35)
is our polynomial, and
              Z    b
                                     b       a                     a+b
                       p (x) dx =                     p (a) + 4p          + p (b)          (36)
               a                         6                          2
Here, A, B and C are constants characterizing the polynomial, and [a; b] is just
some …nite speci…ed interval.
    Simpson’ rule follows by simply treating a function as a concatenation of
quadratic polynomials. The result, typically, will be a continuous function that
approximates f crudely, and if the intervals [a; b] over which a quadratic ap-
proximation is made are fairly small and f is continuous on [a; b], the quadratic
concatenation will …t f su¢ ciently well that we will obtain a good approximation
to the integral of f over some closed interval. Note that this type of approxi-
mation does not necessarily approximate f well: f might be approximated over
the small interval by a fairly high degree polynomial.
    This approximation can be stated explicitly. We refer you to Theorem 4.18
on page 312. When f has continuous fourth derivatives, an error estimate can
be obtained for Simpson’ rule. If we split up the interval of integration (a
…nite closed interval) into n parts, then the error is seen to be proportional
to 1=n4 and the absolute value of the maximum of the fourth derivative over
the interval and the length of the interval. The constant of proportionality is
1=180, which is not a large number. Thus, Simpson’ rule (without modi…cation
to make it computationally more e¢ cient) should converge rather rapidly as n
increases (ignoring round-o¤ errors). In terms of computational e¢ ciency, if T
is a typical time to compute the function values, then the computational time is
on the order of 2nT , which, being proportional to both n and T , is generally not
very fast. However, for simple situations, this is not likely to pose a problem.

   Review: Wallis’formula
   Wallis’ formula is presented on page 536.                     It is an expression which gives
the value of the integral
                                         Z       =2
                                                      cosn xdx                             (37)
   for n     2. The manner in which it is stated by the book presents the
formula in a startling way. One of the curiosities of the formula is that when n

is even, an extra factor of =2 appears, while when n is odd, it does not. We
can present the formula in a more compact way:
                       Z    =2
                                               (n     1)!!
                                 cosn xdx =                E (n)         (38)
                        0                           n!!
    where E (n) takes the value one for n odd, and =2 for n even. The notation
n!! signi…es a multiplication:

                                  n!! = n (n        2) :::               (39)
   where we take products of all numbers from one to n skipping every other
number and including n. Thus,

                                    5!! = 5 3 1

                                    6!! = 6 4 2
   Every such formula like this cannot be regarded as important. However,
Wallis’ formula is at least interesting because it is a general pattern. That
pattern arises from simple integration by parts.


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