Question

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=x^2,x=y^2; about y=−3 V=

Answer 1

The volume V generated by rotating the region about y = −3 is 42π/5 cubic units.

The formula for volume using cylindrical shells is:

V = ∫ 2πrh * dx

In this case, we'll integrate with respect to y, as the curves are defined in terms of y. So, we have:

V = ∫ 2π(y^2 + 3)(y^2 + 3) dy

Determine the limits of integration:

The region is bounded by y = 0 and y = 1, so the limits of integration are:

V = ∫₀¹ 2π(y^2 + 3)(y^2 + 3) dy

Evaluate the integral:

V = 2π ∫₀¹ (y^4 + 6y^2 + 9) dy

V = 2π [(1/5)y^5 + 2y^3 + 9y]₀¹

V = 2π [(1/5) + 2 + 9] = 42π/5