Evaluate the integral

1Z 0

ex+1 x + 1 +

3x + 4 x2 + 3x + 2! dx.

b) Given that F(x) =

1+x2 Z 1+x sint3tant dt, nd F0(1).

2. Determine the type of each of the following improper integrals and determine whether it is convergent or divergent.

a)

Z 0

ex2 5 dx

b)

Z e2

dx xln(lnx)

c)

1Z 0

x esinx 1

dx

d)

Z 1

dx x(lnx)3/2

3. Find the area of the region in the rst quadrant bounded by the parabola y = x2 + 1 and the line y = x + 7

a) by integrating with respect to x (using dx).

b) by integrating with respect to y (using dy).

1

4. Let B be the region bounded by the parabola y = 4xx2 and the line y = 63x. a) Sketch the region B.

b) Express (do not evaluate) the volume of the solid obtained by rotating the region B about the line x = 8 as denite integrals using both the disk and cylindrical shell methods.

c) Express (do not evaluate) the volume of the solid obtained by rotating the region B about the line y = 6 as denite integrals using both the disk and cylindrical shell methods. 5. Let C be the curve y = 2xx2, where 0 x 2. a) Find the length of the curve C.

b) Find the area of the surface obtained by rotating C about the x-axis. c) Find the area of the surface obtained by rotating C about the line y = 1. 6. Let R be the region that is inside the circle r = 3sin and outside the cardioid r = 1 + sin.

a) Find the area of the region R.

b) Find the circumference of the region R.

7. a) Find the center of mass of the wire of 10 meters length that thickens from left to right with density (x) =1 + x 10kg/m. b) Use the Pappus-Guldin Theorem to nd the volume of the surface of revolution obtained by rotating the triangular region in the plane with vertices (0,0), (1,0), and (0,1) about the line x = 2.

8. a) Compute the area of the region that is bounded by the curve with parametric equations x = 3(tsint), y = 3(1cost), where 0 x 6, and the x-axis. b) Compute the volume of the surface of revolution obtained by rotating the portion of the curve with parametric equations x = cos3 t, y = sin3 t that lies in the rst quadrant (0 x 1).

1. a) Evaluate the integral eVEHI 3r + 4 x2 + 3x + 2 dx. 1+22 b) Given that F(x) = / sin (#3) tan (v7) dt, find F'(1). 2. Determine the type of each of the following improper integrals and determine whether it is convergent or divergent. a) b) x In (Inc) C) VI DO d) x(In x )3/2 3. Find the area of the region in the first quadrant bounded by the parabola y = a + 1 and the line a) by integrating with respect to a (using do). b) by integrating with respect to y (using dy).4. Let B be the region bounded by the parabola y = 4x -2" and the line y =6 - 3r. a) Sketch the region B. b) Express (do not evaluate) the volume of the solid obtained by rotating the region B about the line r = 8 as definite integrals using both the disk and cylindrical shell methods. c) Express (do not evaluate) the volume of the solid obtained by rotating the region B about the line y = 6 as definite integrals using both the disk and cylindrical shell methods. 5. Let C be the curve y = v2x - 2, where 0