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
- 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)
d) e)
- (10 pt) Consider the integral
.
- Evaluate this integral.
- Use Simpson’s Rule with to estimate
. What is the maximum error?
- (12 pt) Consider the polar curve
- 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
.
- 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
with error less than .001?
- (15 pt) Determine if the following series converge or diverge.
a) b) c)
- (8 pt) Find the interval of convergence of the power series
- (7 pt) Find all values of for which the series converges. Can you write this as a more familiar function for values of
that makes this series converge?
USEFUL FORMULAS: