009B Sample Final 1, Problem 5

Consider the solid obtained by rotating the area bounded by the following three functions about the $y$-axis:

$x=0$, $y=e^{x}$, and $y=ex$.

a) Sketch the region bounded by the given three functions. Find the intersection point of the two functions:

$y=e^{x}$ and $y=ex$. (There is only one.)

b) Set up the integral for the volume of the solid.

c) Find the volume of the solid by computing the integral.

Foundations:
Review volumes of revolutions

Solution:

(a)

Step 1:
First, we sketch the region bounded by the three functions.
Insert graph here.

Step 2:
Setting the equations equal, we have $e^{x}=ex$.
We get one intersection point, which is $(1,e)$.
This intersection point can be seen in the graph shown in Step 1.

(b)

Step 1:
We proceed using cylindrical shells. The radius of the shells is given by $r=x$.
The height of the shells is given by $h=e^{x}-ex$.

Step 2:
So, the volume of the solid is $\int _{0}^{1}2\pi x(e^{x}-ex)~dx$.

(c)

Step 1:

Step 2:

Final Answer:
(a) $(1,e)$ (See (a) Step 1 for the graph)
(b) $\int _{0}^{1}2\pi x(e^{x}-ex)~dx$
(c)

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