EXAMPLE 4 The region R enclosed by the curves y = 2x and y = x2 is rotated about the x-axis. Find the volume of the resulting solid. SOLUTION The curves y = 2x and y = x2 intersect at the points (o, o y = 2x ) and ( 2, 10 x ). The region between them, the solid of rotation, and a cross section perpendicular to the x-axis are shown in the figures. A cross-section in the plane Py has the shape of a washer ( an annular ring) with inner radius x and an outer radius 2x , so we find the cross- sectional area by subtracting the area of the inner circle from the area of the outer circle: y =x A(x) = 47x2 - T (x2)2 = 1 (2x) 2 - (x 2 ) 2 Therefore, we have V = A(x) dx = 7 ( 4x2 - x" ) dx Ars 64n A(X) 9:37 P