The region between the graph of y = x2 and y = 3x is rotated around the line y = 9.
The volume of the resulting solid is:
Here is a graph of the region:
Using the “Method of Washers”, a vertical slice is taken through the shaded region.
When that slice is rotated about the line y = 9, it will form a washer.
The inside radius of that washer will be 9 – 3x.
The outside radius will be 9 – x2.
The area of the washer is:
A = π(outside radius)2 – π(inside radius)2
A = π(9 – x2)2 – π(9 – 3x)2
The volume of the slice is then:
dV = A dx = [π(9 – x2)2 – π(9 – 3x)2]dx
dV = [π(81 - 18x2 + x4) – π(81 – 54x + 9x2)] dx
dV = π(x4 – 27x2 + 54x) dx
The volume of the solid is then:
3
∫ dV = ∫ π ( x
0
4
− 27x 2 + 54x ) dx
3
#1
&
= π % x 5 − 9x 3 + 27x 2 (
$5
'0
#1 5
3
2&
= π % (3) − 9 (3) + 27 (3) (
$5
'
=
486
π
10
≈ 152.6184