Question

Let r be the region bounded by the curve y = 1 - x², x = 0, and y = 0. Use the shell method to find the volume of the solid generated when r is revolved about the y-axis.

Answer 1

To find the volume of the solid generated when the region bounded by the curve y = 1 - x², x = 0, and y = 0 is revolved about the y-axis using the shell method, set up the integral with respect to y. Evaluate the integral to find the volume.

Step-by-step explanation:

To find the volume of the solid generated when the region bounded by the curve y = 1 - x², x = 0, and y = 0 is revolved about the y-axis using the shell method, we need to set up the integral with respect to y.

The shell method formula for volume is V = 2π * ∫(radius * height * thickness) dy.

In this case, the radius is x and the height is 1 - x². Since we are revolving around the y-axis, x = y and the integral becomes V = 2π * ∫(y * (1 - y²)) dy.

We need to determine the limits of integration. The curve y = 1 - x² intersects the x-axis when y = 0, so the lower limit is 0. The curve intersects the y-axis when x = 0, so the upper limit is 1.

Now we can evaluate the integral: V = 2π * ∫(y - y³) dy from 0 to 1. This can be simplified to V = 2π * (1/2 - 1/4) = π/2.