MATH 161
SPRING 1999
FINAL EXAM
(20 pt) Consider the functions
and
. Write integrals to evaluate the following and evaluate each numerically.
- Find the area between the curves
and
.
- Find the length along the curves
and
from the points
to ![](Image700.gif)
- Find the volume of the solid of revolution obtained by spinning the area between these curves about the x-axis.
- What is the surface area of this solid?
- (20 pt) Evaluate the following integrals:
a)
b)
(
) c) ![](Image704.gif)
d)
e)
![](Image707.gif)
- (10 pt) Consider the integral
![](Image708.gif)
.
- Evaluate this integral.
- Use Simpson’s Rule with
to estimate ![](Image710.gif)
. What is the maximum error?
- (12 pt) Consider the polar curve
![](Image711.gif)
- Find the length of this curve.
- Find the area enclosed by this curve.
- Sketch the curve.
- (6 pt) Sketch the curve with the parametric equations
.
- (12 pt) Consider the function
![](Image713.gif)
.
- Find the Taylor series about the point
.
- Plug in 3 for
and show that
.
- How many terms do we need so that we have using the series above gives us a value for
![](Image717.gif)
with error less than .001?
- (15 pt) Determine if the following series converge or diverge.
a)
b)
c) ![](Image720.gif)
- (8 pt) Find the interval of convergence of the power series
![](Image721.gif)
- (7 pt) Find all values of
for which the series
converges. Can you write this as a more familiar function for values of ![](Image724.gif)
that makes this series converge?
USEFUL FORMULAS:
![](Image725.gif)
![](Image726.gif)