Question

Define q as the region bounded by the functions u(y)=y³-2y+1 and v(y)=1 between y=1 and y=2. If q is rotated around the y-axis, what is the volume of the resulting solid?

Answer 1

The volume of the resulting solid is 0.

Step-by-step explanation:

To find the volume of the solid formed by rotating the region q around the y-axis, we can use the method of cylindrical shells. The volume of each cylindrical shell is given by the formula V = 2πrhΔy, where r is the radius (which is the distance from the y-axis to the function), h is the height (which is the difference between the two functions), and Δy is a small change in y. Integrating this formula over the range of y values from 1 to 2 will give us the total volume.

In this case, the radius r is 1 (since the functions are at a distance of 1 from the y-axis), the height h is given by v(y) - u(y), which simplifies to 0, and Δy is the infinitesimal change in y. Therefore, the integral will evaluate to 0, and the volume of the resulting solid is 0.