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Math 152 “FREE” Final Exam Review Disclaimer: This review by no means covers every concept covered in Math 152 and should not be used as your only source of study. It is a highlight of some of the material. Solutions will be posted at www.math.tamu.edu/∼baurispa/math152/homepage.html 1. Integrate the following. x dx x−9 Z 8x − 7 (b) dx (x − 1)(x2 + 4) Z 1 √ (c) dx 2 x x2 − 4 Z (a) √ Z sec7 θ tan3 θ dθ Z sin2 (ln x) dx x Z x7 cos(x4 ) dx (d) (e) (f) Z ∞ (g) 0 3 1 − 3x + 4 x + 7 dx 2. Set up integrals to find the following. (a) The area bounded by the curves y = x3 , y = −x3 + 2, x = 0, and x = 6. (b) The volume of the solid found by rotating the region bounded by y = 4 − x2 and y = 3 about the line x = 4. (c) The volume of the solid found by rotating the region bounded by y = x2 and y = 2x + 3 about the line y = −2. (d) The volume of the solid whose base is the triangular region with vertices (0, 0), (3, 0), and (0, 4) and whose cross sections perpendicular to the y-axis are semicircles. (e) The length of the curve x = y 3 − y from y = 2 to y = 3. (f) The surface area of the solid found by rotating the curve y = √ x, 0 ≤ x ≤ 9 about the x-axis. (g) The surface area of the solid found by rotating the curve y = t3 , x = t2 , 1 ≤ t ≤ 3 about the y-axis. 3. Find the average value of the function f (x) = tan x on the interval [0, π4 ]. 4. A spring has a natural length of 3 ft. A force of 30 lbs is required to hold the spring stretched to a length of 5 ft. How much work is required to stretch the spring from 4 ft to 8 ft? 5. A trough is 20 m long and has semicicular ends with height 9 m. The trough has a 2 m spout at the top and is full of water. Set up an integral to find the work necessary to pump all the water out of the spout. 1 6. Determine whether the following series converge or diverge. Justify your answer. If the series converges, does it also converge absolutely? (a) (b) (c) (d) (e) ∞ X √ n=1 ∞ X n2 n7 + 12 (2n)! (n!)2 n=0 ∞ X (−1)n √ 3 n=0 ∞ X n2 + 5 (−1)n+1 n ln n n=2 ∞ X ln n n=3 n2 7. Find the sums of the following series, if possible. (a) (b) (c) ∞ X n=3 ∞ X 2 2 − n+1 n 3(−2)n+1 5n n=2 ∞ X (−1)n+1 42n+1 n=0 32n (2n)! 8. Find the radius and interval of convergence for the series ∞ X 32n (3x − 1)n √ n=3 n 9. Express the following functions as Maclaurin series. 7x 5 − x3 (b) x sin(3x5 ) (a) (c) x4 ln(1 + 4x2 ) Z 10. Integrate the following as a power series. 6 x4 e−x dx 11. Find the Taylor series for the function f (x) = 1 centered at x = 1. (7x + 1)2 12. Consider the triangle formed by the vertices P (−1, −5, 2), Q(−4, −2, 1), and R(−1, 0, 2). (a) Find tha angle at vertex Q. (b) Find the area of this triangle. (c) Find a unit vector perpendicular to this triangle. 13. Find the volume of the parallelipiped determined by the vectors < 1, 0, 2 >, < 0, 2, 3 >, and < 8, −5, 6 >. 2