NAME: SCORE: 250 Points (An answer, even if "correct" but lacking work, will NOT receive full credit!) 1. Evaluate each integral by making an appropriate substitution. (a) X dx + 4 flex - ex (b ) Joe tox dx (c) | tan xsec xdx 2. Find the area of the region bounded above by the graph of g(x) = 1 and below by the graph of f (x) =x2 + 2x -2.3. Let R be the closed region between the graphs of y = 2vx and y = 2x on the interval [0, 1]. Find the volume V of the solid obtained by revolving R about the x- axis. ty ...4. Let R be the closed region between the graphs of y = 2vx and y = 2x on the interval [0, 1]. Find the volume V of the solid obtained by revolving R about the y- axis. (Same graph as for Problem3) 5. Find the length of arc on the graph of f(x) = x3/2 from x =1 to x =4 6. Find the area of the surface generated by revolving about the x-axis the curve f(x) = =x' on [0,2]. 7. Fill in the values of f-1(x) and (f-1)'(x) for x = 1, 2,3. Provide detail to support your answers. x f ( x) f'(x) f- 1 (xx) (f- 1)'(x) 1 2 5 2 3 -1 3 1 3 8. Calculate the derivative for y = arctan(Vx2 + 1)9. Use the denition of an improper integral to evaluate the following integrals. Ifan integral converges, evaluate its value. 10. Using the indicated techniques to evaluate the following integrals. Show work detail to support your solutions. Solving using other methods or with no detail is not acceptable. dx (a) I \/14x2 (Trigonometric substitution) 2 (b) I xdx (Trigonometric substitution) 416363 (0) Ixz ln xdx (Integration by Parts) 11. Complete the square in the denominator, make appropriate substitution, and integrate. 1 ldz x x +6x+10 5 12. Find the partial fraction decomposition for the rational function and (u -2)(2 + 3) 5 then evaluate the integral } du (u - 2)(1 + 3) 13. State whether the sequence converges or diverges. If it converges, find its limit. ( n+ 3) 2 - 12 , n = 1,2. ... n 14. Using the Integral Test to test the following series for convergence. Solving using other methods or with no detail is not acceptable. E 1=3 n(Inn)215. Determine whether the following series converge or diverge. Indicate the test you use. 1+ sin k (a) 12 k-1 00 4n -1 ( b ) n=in + 2n--2 16. Determine whether the series converges or diverges. Ifit converges, find its sum. 17. Determine whether the series . (-1)" I'M : converges conditionally, or converges n=0 Vn +3 absolutely, or diverges and give reasons for your conclusions. 18. Find the interval of convergence for the power series _ 1 - x" non+2 19. Use substitution method and a known power series to find power series for -1 Please express your answer in one sigma notation