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) ![](Image156.gif)
(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:
![](Image160.gif)
![](Image161.gif)
(10 pt) Consider the following parametric equation:
![](Image162.gif)
![](Image163.gif)
- 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
![](Image164.gif)
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)![](Image169.gif)
- (10 pt) a) Find the Taylor series about
of ![](Image171.gif)
. What is the interval of convergence?
b) Use the Maclaurin series for
to find
with error less than .001.
USEFUL FORMULAS:
![](Image174.gif)
or ![](Image175.gif)
![](Image176.gif)