MATH 161
FINAL EXAM
(10 pt) a) Find the area between the curves and .
b) Find the volume of obtained by revolving the region bounded by these two curves about the line .
(5 pt) Let be a continuous odd function (). Find the average value of on the interval [-a, a].
(15 pt) Evaluate the following integrals:
a) b) c)
(10 pt) a) Use Simpson’s Rule with n=4 to estimate .
b) Estimate the error for using the midpoint rule and Simpson’s Rule with n=4 to approximate .
(5 pt) Find the arclength along from (0,0) to (4,8).
(10 pt) Solve the differential equation:
(10 pt) Consider the following parametric equation:
- Sketch this ellipse and find its area.
- Write an integral that expresses the surface area obtained by spinning the upper half of this ellipse about the x-axis. After you have done this, let
and solve the integral.
- (10 pt) Sketch the graph of the polar equation and find its length from 0 to p
. (Hint: )
- (15 pt) Determine of the following series converge or diverge.
a) b) c)
- (10 pt) a) Find the Taylor series about of
. What is the interval of convergence?
b) Use the Maclaurin series for to find with error less than .001.
USEFUL FORMULAS:
or