Hello, silencecloak!

Your limits are correct . . .

Rotate around the x-axis, using the cylindrical shell method.

The line has intercepts (1,0) and (0,1).

The parabola opens to the left with vertex at (5,2).

They intersect at (5,0) and (0,5).

The formula is: .

The radius is

The height is: .

Then: .

Got it?