Volumes by Washers

Find the volume of the area between y² = 8x and x = 2 when this area is rotated about the y-axis:

Graph both functions in your calculator or on graph paper to see what the area looks like.
Sketch the figure obtained by rotating the enclosed area around the y-axis.
Draw a representative disk cross section on this solid. Then draw a disk representing the empty space in the middle.
Find an expression for the radius of each disk.

r = 2

r = x = y²/3

The volume of the washer is obtained by subtracting the small disk from the large disk.
To find the total volume, add up all the washer volumes in the solid. (V = p r_L² h - p r_S²h)

(r_L is the radius of the large disk and r_S is the radius of the small disk)

Substitute x = 2 into the original equation to see what the limits on y are.
Remember, this can be done by using a Riemann Sum or calculating an integral.

4
V(x) = p ò (2)² - (y²/8)² dy
-4

4
V(x) = p ò (4 - y⁴/64) dy
-4

                                      | 4
         = p (4y - y⁵/320) |       = p [16 - 1024/320 - (-16 - (-1024)/320)]
                                      | -4

= p (32 -2048/320) = 80.425 cubic units