Question

Find the volume of the solid obtained by rotating the region bounded by x=2-y^2 and x=y about the line x=5. Round to the nearest thousandth.

Answer 1

To find the volume of the solid obtained by rotating the region bounded by x=2-y^2 and x=y about the line x=5, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by x=2-y^2 and x=y about the line x=5, we can use the method of cylindrical shells.

1. First, sketch the region and the axis of rotation to visualize the shape.
2. Set up the integral to find the volume using the formula V = ∫(2πy)(x_2 - x_1)dy, where x_1 and x_2 are the respective x-coordinates of the region's boundaries.
3. Evaluate the integral by plugging in the values of x_1, x_2, and the limits of integration for y.
4. Round the result to the nearest thousandth to obtain the final volume.