112k views
5 votes
Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis. y = x3, y = 8, x = 0; about x = 9 V =

User SimpleApp
by
7.6k points

1 Answer

6 votes

For each
x in the interval
0\le x\le2, the corresponding shell has radius
x+9 (horizontal distance from
x to the rotation axis) and height
8-x^3 (vertical distance between
y=8 and
y=x^3). A shell of radius
r and height
h has area
2\pi rh, so the volume is


\displaystyle2\pi\int_0^2(x+9)(8-x^3)\,\mathrm dx=2\pi\int_0^272+8x-9x^3-x^4\,\mathrm dx=\boxed{\frac{1176\pi}5}

User Nikit
by
8.9k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.