Question

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. y = x3/2, y = 8, x = 0

Answer 1

Solve for the value of x such that f(x)=8
imma assume it is y=x^(3/2)
so
solve 8=x^(3/2)
x=4
so integrate from x=0 to x=4
$\pi \int\limits^4_0 {(x^ \frac{3}{2})^2 } \, dx$ is the area
$\pi \int\limits^4_0 {x^3 } \, dx$=
$\pi [ \frac{1}{4}x^4 ]^4_0$=
$\pi (4^3)$=
64π cubic units