Question

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. x =

6 − y, y = 0, x = 0; about the y-axis

Answer 1

To find the volume of the solid obtained by rotating the given curves about the y-axis, we can use the method of cylindrical shells. The integral for the volume is V = ∫(2π(6 - y)Δy).

Step-by-step explanation:

To set up the integral for the volume of the solid obtained by rotating the region bounded by the curves x = 6 - y, y = 0, x = 0 about the y-axis, we need to use the method of cylindrical shells.

The volume of each shell can be represented by the circumference of the shell multiplied by its height. The circumference is given by 2πr, where r is the distance from the y-axis to the shell. The height is given by Δy, which is the change in y value along the curve. The integral for the volume is then:

V = ∫(2πrΔy)

To find r in terms of y, we need to solve the equation x = 6 - y for x. This gives us x = 6 - y. Thus, r = x = 6 - y.

Therefore, the integral for the volume is:

V = ∫(2π(6 - y)Δy)