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Math 126 Carter Test 1 Spring 2013 General Instructions: Write your name on only the outside of your blue book. Do all of your work and write your answers inside your blue book. Put your test paper inside your blue book as you leave. Solve each of the following problems; point values are indicated on the problems. There are 114 total points available on the test. Slices of ripe avocado are a pleasant addition to a sandwich. 1. (8 points each) Compute the following: (a) Z (4x3 − 6x2 − 8x + 4) dx (b) Z π sin (θ) dθ 0 (c) Z 32 x−6/5 dx 1 (d) Z sin (4t) cos3 (4t) dt (e) Z 4 2 dx x(ln (x))2 (f) Z sin (x)ecos (x) dx (g) Z 2y dy y 2 + 16 (h) Z 1 5−t dt 0 2. (5 points) Complete the statement of the fundamental theorem of calculus that begins: “Assume that f (x) is a continuous function on the open interval (a,b), and let F (x) be an anti-derivative of f (x) on the closed interval [a, b], then [. . .] .” 3. (5 points) Use the technique of proof by induction, to show that N X j= j=1 (Hint: factor, don’t multiply.) 1 N (N + 1) . 2 4. (10 points) Compute the volume of the cone that is obtained by rotating the triangle that is bounded by y = 0, x = 0 and y = 10 − x/3 about the x-axis. 5. (10 points) Compute the volume that is obtained by rotating the area under the parabola y = x2 between x = 2 and x = 5 about the y-axis. 6. (10 points) √ Compute the surface area of the sphere that is obtained by revolving the curve y = 25 − x2 between x = −5 to x = 5 about the x-axis. Recall, that the surface area formula is given by A= Z b q 2πf (x) 1 + (f 0 (x))2 dx. a (Hint: the resulting integral is easy to calculate.) 7. (10 points) If a force of 90N ewtons stretches a spring 1 meter beyond its natural length, how much work does it take to stretch the spring 5 meters beyond its natural length? Recall that 1 Newton = 1 kilogram meter per square second, and that Hook’s law states that the force required to stretch a spring is proportional to the length that it is stretched. 2