Math 152 - Spring 2008
Challenge for Exam 2 (this is not a substitute for recommended homework!)
1. Determine whether an integral is convergent or divergent. If possible, evaluate the integral.
Comment: For some of the integrals you should be able to find an antiderivative and use it to
make a conclusion; for others you may need to use the comparison theorem. You may also need
to find a linear approximation to a function near a point x0 : f (x) ≈ f (x0 ) + f 0 (x0 )(x − x0 ) )
Z 2
1
•
dx
1 ln x
√
Z 2/π
1
1
•
cos 2 dx
3
x
x
0
Z 1 cos 1
x dx
√
•
3
x
0
Z π/2
tan x
dx
•
x2
1
Z ∞
arctan x
•
dx
x2
Z1 ∞
dx
•
(to practice with partial fractions, find an antiderivative first).
2
2 2
1 x (1 + x )
Z ∞ 3
x − 2x2 + x + 1
•
dx (to practice with partial fractions, find an antiderivative first).
x4 + 5x2 + 4
−∞
Z 1
•
ln2 x dx
0
2. Find the volume of the solid of revolution obtained by revolving the region bounded by
1
, x = −1, x = 1, y = 0 about the x−axis.
y=
1 + x2
Z 1
3. Show that for any positive integer n,
(ln x)n dx = (−1)n n! (use the method of mathemat0
ical induction)
Z
4. a) Show that for a > 0 the integral Γ(a) =
∞
e−x xa−1 dx converges. (this integral defines
0
so called Gamma function, Γ(a))
b) Compute Γ(1).
c) Show that Γ(a + 1) = aΓ(a) for any a > 0.
d) Use c) to show that if a = n is a positive integer, then Γ(n) = n! where ( n! = 1 · 2 · ... · n)
Z ∞
5. The Laplace transform of a continuous function f is defined as F (s) =
f (t)e−st dt.
0
a) Suppose that for t > 0 a function f is nonnegative and grows slower than some exponential
functionRM eat (0 ≤ f (t) ≤ M eat for some constants M, a). Show that in this case the integral
∞
F (s) = 0 f (t)e−st dt converges if s > a, and thus f has the Laplace transform F (s), s > a.
b) Show that the following functions satisfy the condition of convergence given in a) and then
find their Laplace transforms: i) f (t) = 1, ii) f (t) = e5t , iii) f (t) = t, iv) f (t) = sin t .
Z h
f (x) dx if f (x)
6. Show that the Simpson’s formula gives the exact value of an integral
−h
is a polynomial of degree less or equal to three. Hint: let f (x) = Ax3 + Bx2 + Cx + D.
Apply Simpson’s formula to two intervals with x0 = −h, x1 = 0, x2 = h and show that both
2
Simpson’s formula and direct evaluation of f on [−h, h] give the same result, Bh3 + 2Dh.
3
1
7. a) Show that the region R = (x, y)| 0 ≤ y ≤ , x ≥ 1 has infinite area.
x
b) Show that if we rotate this region about the x−axis, the resulting solid has a finite volume.
c) Show that this solid of revolution has an infinite surface area (for the formula, see 9.4,
page 550). Interesting result: you can fill this solid with paint but you will never have enough
paint for the surface! Such a monster.
Z
1 x
f (t) dt .
8. We define the average value of a function f (x) on (0, ∞) as fave = limx→∞
x 0
Find the average value of f (x) = arctan x on (0, ∞).
9. Solve the equation
1
dy
=
.
dx
x + ey
10. The magnitude of the repulsive force between two charges with the same sign, one of size 1
q
and the other of size q, is F =
where 0 is a constant and r is the distance between
4π0 r2
the charges (all quantities are in SI units). The potential V at a point P due to charge q is
defined as work done in bringing a unit charge to the point P from infinity along the straight
line joining q and P. Find the formula for V . (Assume that the charge q is at the origin and
the point P is at the distance d from it. )
11. pr. 42, 43, 44, page 535
12. pr. 24, 28, 29, 33, page 541
13. pr. 28, page 560