To determine

To find:

The volume of the solid obtained by rotating the region bounded by the given curves about the specified line and sketch the region, the solid, and a typical disk or washer.

Answer

V= 6415π

The sketch of the bounded region, the solid, and a typical washer:

Explanation

1) Concept:

i. If the cross section is a washer with the inner radius rin and the outer radius rout, then the area of washer is obtained by subtracting the area of the inner disk from the area of the outer disk.

A=π outer radius2 -π inner radius2

ii. The volume of the solid revolution about y-axis is

V= ∫abA(y)dy

2) Given:

y2=x,  x=2y ; about the y- axis

3) Calculation:

The region bounded by y2=x, x=2y and the solid obtained by rotation about the y- axis is shown below:

Here, the region rotated about the y – axis, so the cross-section is perpendicular to y-axis.

A cross section of the solid is the washer with the outer radius 2y and the inner radius y2.

So, its cross sectional area is

Ay=πouter radius2-πinner radius2 =π2y2-πy22

Ay=π4y2-y4

The region of integration is bounded by y2=x and x=2y.At the point of intersection of both curves

y2=2y 

y2-2y=0

yy-2=0

y=0  and y=2

The solid lies between  y=0 and y= 2

Therefore, the volume of the solid revolution about the y-axis,

V= ∫02Aydy=∫02π(4y2-y4)dy

By using the fundamental theorem of calculus part 2 and the power rule of integration,

V=π4y33-y5502

By substituting the limits of integration,

=π 4·233-255-4·033-(0)55

=π323-325-0

=π 16015-9615=π6415

V=6415π

Therefore, the volume of the solid obtained by rotating the region bounded by the given curves about the y axis is

V=6415π

Conclusion:

The volume of solid obtained by rotating the region bounded by the given curves about the y axis is V=6415π

Sketch of bounded region, the solid and a typical washer: