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.y = 6x − x2, y = x; about x = 8

Answer 1

To set up the integral for the volume of the solid obtained by rotating the region bounded by the curves y = 6x - x^2 and y = x about the line x = 8, use the method of cylindrical shells.

Step-by-step explanation:

To set up the integral for the volume of the solid obtained by rotating the region bounded by the curves y = 6x - x^2 and y = x about the line x = 8, we can use the method of cylindrical shells.

We need to find the limits of integration along the x-axis by finding the x-values where the two curves intersect. By setting 6x - x^2 = x, we get a quadratic equation that can be solved to find the x-values. Let's call these x-values x_1 and x_2.

The integral for the volume is then given by:

V = ∫[from x_1 to x_2] 2πx(y - 8) dx