1. ( ) Part (a) Shaded area = x dx - e dx1
x dx - e dx ( )
-3x
To find the lower limit of integration, use the graphing calculator to see where the curves intersect.

2. The curves intersect at x = .238734
Shaded area = A 1
x dx - e dx ( )
-3x
.2387
The curves intersect at x =

3. ( ) Shaded area = x dx - e dx Shaded area = .442 or .443 units2 -3x 1
.2387
Shaded area = .442 or .443 units2

4. The rectangles are perpendicular to the axis of revolution.
Part (b)
The rectangles are perpendicular to the axis of revolution.
In addition, they do not physically touch the axis of revolution.
y = 1
We’ll need to use the washer method.

5. Outer radius: R = 1 – e-3x Inner radius: r = 1 – x
y = 1

6. Plugging this into the graphing calculator gives us…
Volume = p
.2387 1
1 - e x ( )
-3x 2
Plugging this into the graphing calculator gives us…
y = 1
Volume = .453p or or 1.424

7. Part (c) 5y y ( )
-3x
x - e ( )
-3x
x - e 5
The volume of this solid is the sum of the areas of all rectangles from x=.2387 to x=1.

8. ( ) ( ) ( ) Volume = = 1.554 x - e 5 5y y x - e x - e 5 2 -3x -3x -3x
.2387 1
x - e ( )
-3x 5 2 5y y ( )
-3x
x - e ( )
-3x
x - e 5