Question

Let r be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when r is revolved about the y-axis. Find the volume of the solid generated when the region bounded by the curves y = y0 and x = x0 in the first quadrant is revolved about the y-axis?

Answer 1

To find the volume of a solid generated by revolving a region around the y-axis, the shell method involves integrating the surface area of cylindrical shells over the defined range. The volume of a sphere is given by 4πr³/3.

Step-by-step explanation:

The student appears to be asking for the volume of a solid generated by revolving a region bounded by given curves around the y-axis. According to the shell method, if we revolve a region bounded by y = y0 and x = x0 in the first quadrant around the y-axis, we can visualize thin cylindrical shells forming the 3D object. To find the volume of such a solid, we integrate the surface area of these cylindrical shells over the range of x values that define the region. Each cylindrical shell has a circumference of 2πx (where x is the distance from the y-axis), a height given by the function y, and a thickness dx.

The volume V of the solid is the integral of the product of the circumference, the height, and the thickness:

V = ∫x0x1 2πx • y(x) • dx

This formula needs the exact functions or values for y(x), x0, and x1 to compute the volume.

If the student is also inquiring about a sphere, the volume of a sphere of radius r is given by (4πr3/3) and not 4πr2 which is actually the surface area.