Area by Parts and Rotational Volume
Practice
5.1 Given:
a) Find the coordinates of the intersection points using your TI.
b) Set up the integrals to find the area enclosed by y1 and y2
c) Evaluate the integrals in part b) using the TI.
Solve on the calculator to get 32.75
5.2 Let R be the region bounded by 
a) Find the area of R
b) Find the volume of R when it is spun about the x-axis
Solve using disk about the x axis:
c) Set up a shell integral for the volume when R is revolved about the y axis. (do not evaluate)
Solve using shell about the y axis:
5.3 Let the region R denote the area bounded by 
, and the x axis.
a) Find the volume when R is rotated around the y-axis.
Solve using shell about the y axis:
b) Why is the disk method not possible around the y-axis?
Because we can’t solve for x. (we are unable to put 
in terms of x)
5.4 Let R denote the region bounded by 
a) Find the area of R
b) Revolve R about the x-axis. Solve for volume.
Solve using washer method about the x-axis.
5.5 Let R denote the region bounded by 
. Set up the integral for R being revolved about the line x = 3. Do not solve.
Solve using shell method about a line:
5.6 Let R equal the region bounded by 
. Set up the integral for R being revolved about the x-axis by shell method. No not solve.
Find points of intersection
, (2)²
.
Solve using shell about the x-axis.
You will need to find the volume of y2 from y = 0 to y = 4 and then the volume of y1 from y = 4 to y = 12.
The area would be denoted by:
Multiplied by 2 because R is symmetric across the y axis
