- (10 pt) For the following two rational functions, show the partial fractions decomposition form.
a)
b) ![](Image749.gif)
![](Image750.gif)
- (20 pt) For the following integrals, tell what technique/substitution that you would use and briefly say why (you do not have to work the integrals out).
a)
(where
is an integer) b)![](Image753.gif)
c)
d) ![](Image755.gif)
- (20 pt) Evaluate the following integrals
a)
b) ![](Image757.gif)
![](Image758.gif)
c)
d) ![](Image760.gif)
- (5 pt) Show that the integral
converges (do not try to find an antiderivative of this).
- (10 pt) Solve the initial value problem
, ![](Image763.gif)
- (15 pt) Use Simpson’s Rule and an appropriate integral to estimate ln(3) with error smaller than .001.
- (20 pt) For the following problem consider the quarter circle bounded by
, and ![](Image765.gif)
in the first quadrant.
- Find the centroid of this region (assuming constant density).
- Find the volume obtained when we revolve this region about the line
![](Image766.gif)
.
- EXTRA CREDIT (10 pt)
- Find a parametric equation that describes the ellipse
![](Image767.gif)
- Find the surface area obtained when the above ellipse is revolved about the x-axis. Check your answer by taking the limit as
. (You may assume that ![](Image769.gif)
)