Final answer:
The volume of the solid formed by rotating the region between y=x and y=x² about the line x=-1 can be found using the disk or washer method, where the integral V = π∫_0^1 [(1+x)² - (1+x²)²] dx provides the solution.
Step-by-step explanation:
To find the volume of the solid formed by rotating the region enclosed by the curves y=x and y=x² about the line x=-1, one typically uses the disk or washer method of integration.
The region of interest lies between the intersections of the curves, which occur at x=0 and x=1.
When rotating about x=-1, we consider the radius of each disk to be the distance from x=-1 to the function value at a given x, which is (1+x) for y=x and (1+x²) for y=x².
To set up the integral for the volume, we subtract the inner function (x²) from the outer function (x) after adjusting their radii due to the axis of rotation being x=-1.
The volume integral is then V = π∫_0^1 [(1+x)² - (1+x²)²] dx. This integral can be solved using standard calculus methods to find the exact volume of the solid.