Math Makeup Exam - 3/14/2018

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1 Math 22 - Makeup Exam - 3/4/28 Name: Section: The following rules apply: This is a closed-book exam. You may not use any books or notes on this exam. For free response questions, you must show all work. Answers without proper justification will not receive full credit. Partial credit will be awarded for significant progress towards the correct answer. Cross off any work that you do not want graded. For multiple choice questions, circle the letter of the best answer. Make sure your circles include just one letter. These problems will be marked as correct or incorrect; partial credit will not be awarded for problems in this section. You have 5 minutes to complete this exam. When time is called, stop writing immediately and turn in your exam to the nearest proctor. You may not use any electronic devices including (but not limited to) calculators, cell phone, or ipods. Using such a device will be considered a violation of the university s academic integrity policy and, at the very least, will result in a grade of for this exam. Problem Points Score Total:

2 Math 22 Make-up Exam - Page 2 of 9 3/4/28 Part I: Free Response. Consider the function f(x) = ln(x). (a) ( points) Set up but do not evaluate a Riemann Sum using right endpoints for the area bounded between f and the x-axis over the interval [, e 2 ]. A = lim x = e2 n x k = a + k x = + e2 n k n n k= ( ln + e2 n k ) e 2 (b) ( points) Set up and evaluate a definite integral to find the area bounded between f and the x-axis over the interval [, e 2 ]. n OPTION : Set up in terms of x. Using integration by parts (u = ln x; dv = dx), we get e 2 OPTION 2: Set up in terms of y. e 2 ln x dx = x ln x x = ( e 2 ln e 2 e 2) ( ln ) = 2e 2 e 2 + = e 2 + A = 2 e 2 e y dy 2 = e 2 y e y = (2e 2 e 2 ) (e 2 () e ) = e 2 +

3 Math 22 Make-up Exam - Page 3 of 9 3/4/28 2. (5 points) At Charlie s Chocolate Factory, a tank in the shape of an inverted right circular cone has a height of meters and a radius (at the top) of 6 meters is filled with chocolate pudding to a height of 2 meters. In order to sterilize the tank, the factory needs to empty the tank. Set up but do not evaluate an integral to find the work required to empty the tank by pumping the chocolate pudding through a hole in the top of the tank. Note: the weight-density of chocolate pudding is 2,78 N/m 3. The volume of a slice of pudding is given by dv = πr 2 dx. We want to express r in terms of x. To do so, we can use similar triangles. r x = 6 = r = 3 5 x So the volume of a slice becomes: dv = 9π 25 x2 dx The slice at height x has to travel x units to get to the top of the tank. W = 2, π 25 x2 ( x) dx

4 Math 22 Make-up Exam - Page 4 of 9 3/4/28 3. ( points) Evaluate 2x 2 + 7x 3 x 3 2x 2 3x dx. We begin by factoring the denominator: The partial fraction decomposition is then 2x 2 + 7x 3 x 3 2x 2 3x = 2x2 + 7x 3 x(x 3)(x + ) 2x 2 + 7x 3 x(x 3)(x + ) = A x + To find the coefficients A, B, C, we solve B x 3 + C x + 2x 2 + 7x 3 = A(x 3)(x + ) + Bx(x + ) + Cx(x 3) Plug in x = : 3 = 3A = A =. Plug in x = 3: 36 = 2B = B = 3. Plug in x = : 8 = 4C = C = 2 Plugging back into our integral, we geometry 2x 2 + 7x 3 x 3 2x 2 3x dx = x + 3 x x + dx = ln x + 3 ln x 3 2 ln x + + C

5 Math 22 Make-up Exam - Page 5 of 9 3/4/28 4. Suppose R is the region enclosed by y = ln x, y = 2, and x =. (a) (4 points) On the plot given below, label all of the curves. Also, label all points at which the curves intersect as ordered pairs (x, y). (b) (6 points) Set up but do not evaluate an integral which represents the area of R by integrating with respect to x. e 2 ( ln x) dx (c) (6 points) Set up but do not evaluate an integral which represents the area of R by integrating with respect to y. 2 (e y ) dy

6 Math 22 Make-up Exam - Page 6 of 9 3/4/28 (d) (8 points) Compute the area of R. 2 (e y ) dx = e y y 2 = e 2 2 ( ) = e 2 3 (e) (6 points) Suppose R is the base of a solid whose cross sections taken perpendicular to the x-axis are squares. Set up but do not evaluate an integral which represents the volume of this solid. e 2 ( ln x) 2 dx

7 Math 22 Make-up Exam - Page 7 of 9 3/4/28 Part II: Multiple Choice 5. (5 points) Which of the following is the arc length of f(x) = 2 3 (x )3/2 from x = to x =? (a) (b) (c) 2 (d) 3/2 (e) 2/3 6. (5 points) Evaluate the following improper integral: (a) x dx. (b) 2 (c) (d) 2 (e) +

8 Math 22 Make-up Exam - Page 8 of 9 3/4/28 7. (5 points) Which of the following functions is a solution to the given initial value problem? dy dx = x 2 y ( ) = π 2 (a) y = sin x + 5π 6 (b) y = sin x + π (c) y = ln ( x) + π (d) y = tan x + π (e) y = tan x (5 points) Suppose F (x) = x f(t) dt, where f(t) is shown below. Which of the following quantities is the greatest? (a) F () (b) F () (c) F (5) (d) F () F (5) (e) F (5) F ()

9 Math 22 Make-up Exam - Page 9 of 9 3/4/28 9. (5 points) Which of the following integrals is/are improper? I. x 2 dx II. x dx III. π/2 tan x dx (a) I only (b) II only (c) I and II only (d) I and III only (e) I, II, and III

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