Hints to the Exercises Integral Calculus

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					                                                     Review
                                  Fundamental Concepts and Techniques of Calculus

                            Hints to the Exercises: Integral Calculus
                                                 Last updated: 041012



1. Integration By Parts:                                             (c) Use the first Pythagorean identity and rewrite

    (a) Use integration by parts (R 4:6c) and set u = x                       cos3 x = cos2 x cos x = (1 − sin2 x) cos x.
        and v = ex .
                                                                            Then apply (R 4:5b).
    (b)
                                                                     (d)
    (c) Use integration by parts (R 4:6c) and set u = x
                                                                     (e) You can use the trigonometric identity cos2 x =
        and v = sin x                                                    1
                                                                         2 (1 + cos 2x) to reduce the power of cos in
    (d) Integrate by parts (R 4:6c) twice such that the                  the integrand or apply the recursion formula
        remaining integral is a multiple of the original                 (R 4:5q).
        integral. Combine those integrals on the left side
                                                                     (f) Use the trigonometric identities
        of the equation.
                                                                                                    1
    (e)                                                                             sin x cos x =   2   sin(2x)
                                                                                            2       1
    (f)                                                                                  cos x =    2   1 + cos(2x)
                                                                                            2       1
    (g)                                                                                  sin x =    2   1 − cos(2x)
    (h)                                                                     to reduce the number of factors.
     (i)                                                             (g)
     (j)                                                             (h)
    (k)                                                               (i)
     (l)                                                              (j)
   (m)                                                               (k)
    (n)                                                               (l)

    (o)                                                             (m)
                                                                     (n)
    (p)
    (q)                                                          3. Substitution:
    (r)                                                              (a) √ the substitution (R 4:6b) given by y = 1 +
                                                                         Try
    (s)                                                                    x.

     (t)                                                             (b)
                                                                     (c)
2. Trigonometric Integrals:
                                                                     (d)
    (a) Use the Second Pythagorean identity and                      (e)
        rewrite                                                      (f)
           tan3 x sec3 x = sec2 x tan2 x(sec x tan x)                (g) Try the substitution (R 4:6b) given by y =
                                                                         sin−1 (2x).
                        = sec2 (sec2 x − 1)(sec x tan x).
                                                                     (h)
           Then apply (R 4:5b).                                       (i)
    (b)                                                               (j)


                                                             1
    (k)                                                           (w)
     (l)                                                           (x)
   (m)                                                             (y)
    (n)                                                        5. Improper Integrals:
    (o)
                                                                   (a)
    (p)
                                                                   (b)
    (q)
                                                                   (c)
     (r)
                                                                   (d)
    (s) Reduce the integrand by the substitution                   (e)
        (R 4:6b) given by y = ex .
                                                                   (f)
     (t) Complete the square and use trigonometric sub-
         stitution (R 4:6(b)iii).                              6. Applications of Integration:
    (u)                                                            (a) Area Between Two Curves:
    (v)                                                                    i.
    (w)                                                                   ii.
    (x)                                                                  iii.
    (y) Use trigonometric substitution (R 4:6(b)iii) and           (b) Volume By Slicing:
        set x = a tan θ.                                                    i.
                    3
    (z) let u = −x                                                         ii.
                                                                         iii.
4. Partial Fractions:
                                                                          iv.
    (a)                                                                    v.
    (b)                                                            (c) Volume By Shells:
    (c)                                                                     i.
    (d)                                                                    ii.
    (e)                                                                  iii.
                                                                          iv.
     (f)
                                                                           v.
    (g)
                                                                   (d) Arc Length:
    (h)
                                                                           i.
     (i)                                                                  ii.
     (j)                                                           (e) Surface Area:
    (k)                                                                     i.
     (l)                                                                   ii.
   (m)                                                                   iii.
    (n)                                                                   iv.
    (o)                                                            (f) Mass From Density:
    (p)                                                                     i.
                                                                           ii.
    (q)
                                                                         iii.
     (r)
                                                                          iv.
    (s)                                                            (g) Center of Mass and Centroid:
     (t)                                                                   i.
    (u)                                                                   ii.
    (v)                                                                  iii.

                                                           2
            iv.                                                   (e) Find the Length of the Parametric Arc:
             v.                                                         i.
            vi.                                                        ii.
           vii.
                                                              8. Polar Coordinates:
          viii.
            ix.                                                   (a) Write the Equation in Polar Coordinates:
    (h) Fluid Pressure:                                                 i.
             i.                                                        ii.
            ii.                                                       iii.
    (i) Work:                                                     (b) Write the Equation in Cartesian Coordinates:
              i.                                                        i.
             ii.                                                       ii.
           iii.                                                       iii.
            iv.                                                   (c) Sketch the Polar Curves:
7. Parametric Curves:                                                    i.
                                                                        ii.
    (a) Express the Parametric Curve by an Equation in
                                                                      iii.
        x and y:
                                                                       iv.
              i.
                                                                        v.
             ii.
                                                                      vi.
           iii.
                                                                  (d) Calculate the Area enclosed by the Polar Curve:
            iv.
             v.                                                          i.
                                                                        ii.
    (b) Find a Parametrization x = (t), y = y(t),
        t ∈ [0, 1] for                                                iii.
              i.                                                       iv.
             ii.                                                        v.
           iii.                                                   (e) Find the Slope of the Polar Curve:
            iv.                                                         i.
    (c)                                                                ii.
    (d) Find the slope of the given curve at the given                iii.
        point and give an equation of the tangent line:           (f) Find the Length of the Polar Curve:
             i.                                                         i.
            ii.                                                        ii.




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