MAT 137Y: Calculus! Problem Set C. This problem set in intended to help you prepare for Test #3. It is not comprehensive: it only contains some problems that were not included in past problem sets or in past tutorials. You do not need to turn in any of these problems. 1. In this question, we study x x + 1dx. Z I= (a) Evaluate I using the change of variable formula. (b) Evaluate I using integration by part. (c) Explain your results. 2. We have a ball of radius R. We drill a cylindrical hole of radius r < R and of length h < 2R in the ball. What is the volume of the ball with the hole drilled in it? 3. Evaluate: (a) Z (b) ex dx 9 e2x 2 Z x2x dx 0 (c) 3x5 6x2 dx x3 1 Z (d) 1 Z 0 (e) Z e x dx x x2 dx 1x (f) Z /3 tan x sec3/2 xdx 0 4. Prove that Z lim a0+ 0 a 1 dx = 0. x 5. Let f and g be two continuous function on R such that f (x) > g(x) for all x R. Prove that the area of the domain bellow y = f (x), above y = g(x), left of x = 2 and right of x = a goes to zero when a 2 . 6. Does the following improper integrals converge? (a) x dx (1 + x2 )2 Z Z 1 (b) (Hint: on [e, ), lnx < e dx x ln x x) (c) Z 4 x3 arctan x + x sin x dx x5 + ex + 2 7. Evaluate Z 4 3x2 dx 1 using a Riemann Sum with a regular partition. 8. Let f, g, h be continuous, differentiable functions with continuous derivatives. Suppose d g (h(y)) f 0 (y) = dy [f (y)g (h(y))] Evaluate Z f (j(x))g 0 (h(j(x))h0 (j(x))j 0 (x)dx (1)