We need to find the volume when the region bounded by y equals x squared and y equals x is rotated around the line y equals x. First, let's find where these curves intersect by setting x squared equal to x.
Solving x squared equals x, we get x squared minus x equals zero, which factors as x times x minus one equals zero. This gives us x equals zero or x equals one, so the intersection points are at origin and at one comma one.
Now we need to set up the volume calculation. Since we're rotating around the line y equals x, we need to find the distance from any point in our region to this rotation axis.
The distance from a point x comma y to the line y equals x is given by the point-to-line distance formula. This gives us d equals the absolute value of x minus y divided by square root of two. Since our region satisfies y less than or equal to x, we can remove the absolute value signs.
Now we set up the volume integral using Pappus's theorem. The volume equals the double integral over the region of two pi times the distance times the area element d A.
Substituting our distance formula, we get the volume as square root of two pi times the double integral from zero to one and from x squared to x of x minus y d y d x. This represents integrating over vertical strips in our region.