1. Evaluate each integral by making an appropriate substitution. x x 2+4 dx (a) x x (b) dx 0 ee xe +e x (c) tan x sec4 xdx 1 2. Find the area of the region bounded above by the graph of g( x)=1 2 and below by the graph of f (x )=x +2 x2 . 3. Let R be the closed region between the graphs of y=2 x and y=2 x Find the volume V of the solid obtained by revolving R about the x- axis. on the interval [0, 1]. 4. Let R be the closed region between the graphs of y=2 x and y=2 x 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) 2 f (x )= x 3/2 3 5. Find the length of arc on the graph of from x=1 to x=4 6. Find the area of the surface generated by revolving about the x-axis the curve 1 f (x )= x 3 3 on [0,2] . ' 7. Fill in the values of f 1 ( x) and ( f 1 ) (x) for support your answers. x f (x) 1 2 3 2 3 1 ' f (x ) 5 1 f 1 x=1, 2,3 . Provide detail to ' ( x) ( f 1 ) (x) 3 8. Calculate the derivative for y=arctan ( x 2 +1) 9. Use the definition of an improper integral to evaluate the following integrals. If an integral converges, evaluate its value. 1 1 0 x0 .9 dx (a) (b) 1 2x dx x 2 +1 2 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. (a) dx 14 x 2 (Trigonometric substitution) 2 x 2 dx 16x (b) (Trigonometric substitution) x2 ln xdx (c) (Integration by Parts) 11. Complete the square in the denominator, make appropriate substitution, and integrate. 1 x 2 +6 x+10 dx 5 12. Find the partial fraction decomposition for the rational function (u2 )(u+3 ) and 5 (u2)(u+3 ) du then evaluate the integral 13. State whether the sequence converges or diverges. If it converges, find its limit. { (n+3)2 n 2 , 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. 1 n(ln n)2 n=3 3 15. Determine whether the following series converge or diverge. Indicate the test you use. 2 1+sin k k =2 k1 (a) (b) 4 n1 n3+2 n22 n=1 n 1 3 ( ) ( ) n=0 3 4 16. Determine whether the series converges, find its sum. [ n converges or diverges. If it (1)n n=0 n+3 17. Determine whether the series converges conditionally, or converges absolutely, or diverges and give reasons for your conclusions. 1 18. Find the interval of convergence for the power series xn n=0 n+2 3x e 1 x 19. Use substitution method and a known power series to find power series for . Please express your answer in one sigma notation. 4