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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