Question

Calc II 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
x = 4y^2 - y^3
x = 0

Answer 1

The volume of the solid obtained by rotating the region bounded by the curves x = 4y^2 - y^3 and x = 0 about the x-axis using the method of cylindrical shells is given by the integral:

V = 2π ∫[0,1] y(4y^2 - y^3) dy

Simplifying the integrand, we get:

V = 2π ∫[0,1] (4y^3 - y^4) dy

Integrating, we get:

V = 2π [(y^4 - (1/5)y^5)]|[0,1]

V = 2π [(1 - (1/5))] = (8/5)π

Therefore, the volume of the solid is (8/5)π cubic units.