Example 4 The region 9 enclosed by the curves y = 2x and y = x 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 (0, ) and (2, . The region between them, the solid of rotation, and a cross section perpendicular to the x-axis are shown in the figures below. y = 2x y =x Graph Graph Description A(X ) A cross-section in the plane Px has the shape of a washer (an annular ring) with inner radius and an outer radius so we find the cross-sectional area by subtracting the area of the inner circle from the area of the outer circle. A(x) = 4xx2 - *(x2)2 = x Therefore, we have V = 10 "A( x ) ax = 60 * ( 4x 2 - * " ) dx Need Help? Read It2. [-/1 Points] DETAILS SCALCET9 6.2.011. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x + 1, y = 0, x = 0, x = 2; about the x-axis V = Sketch the region. 3 w 2 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 Sketch the solid and a typical disk or washer. O Interactive 3D Graph Help . NW -5 -4 -3 -2 -1- 2 3 4 5 -3 --4 - 5 O Interactive 3D Graph Help Interactive 3D Graph Help Interactive 3D Graph Help 5 -4 5 5 - -5 -4 -3 -2 -1 N -5 -4 -3 -2 -1 -7 2 -5 -4 -3 -2 -1 3 5 Need Help? Read It Submit Answer Viewing Saved Work Revert to Last Response3. [-/1 Points] DETAILS SCALCET9 6.2.013. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = Vx - 1, y = 0, x = 4; about the x-axis Sketch the region. w N Sketch the solid and a typical disk or washer. O Interactive 3D Graph Help 10 - -10 10 --5 -10 O C O Interactive 3D Graph Help Interactive 3D Graph Help .on Interactive 3D Graph Help 10- 10- 10- -10 -5 5 10 -10 10 - -5 - -5 --10 -10 --10 Need Help? Read It Watch It