Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

Question

x=3y^2, x=3; about x=3

Answer 1

We set the given curves equal to each other to find the bounds of integration, which are + and -1. Then, we use the formula for rotating a curve about an axis (x=3 in this case) so the integrand is the function (3-3y^2)^2 because we are integrating with respect to y.. Then, we simply integrate leaving a constant of pi in front of the integral.

Answer 2

We can shift the region left by 3 units and rotate it about the y-axis.

The shifted region is between x = 3y² - 3 and x = 0.

The disk area is π(3-3y²)² and the integration domain is y∈(-1,1).

V = ∫_-1¹ π(3 - 3y²)² dy = ∫_-1¹ π(9y⁴ - 18y² + 9) dy = π(9y⁵/5 - 6y³ + 9y)|_-1¹ = 48π/5