#### To determine

**To find:**

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

#### Answer

V=24845π

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 the 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 of revolution about y-axis is

V= ∫abA(y)dy

**2) Given:**

x=2-y2, x=y4; about the y- axis

**3) Calculation:**

The region bounded by x=2-y2, x=y4 and solid obtained through rotation about the y- axis are shown below:

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

A cross section of the solid is the washer with the outer radius 2-y2 and the inner radius y4

So, its cross sectional area is

Ay=πouter radius2-πinner radius2 =π2-y22-πy42

Ay=π2-y22-y8=π(4-4y2+y4-y8)

The region of integration is bounded by x=2-y2, x=y4.

At points of intersection,

y4=2-y2,

y4+y2-2=0,

y4+2y2-y2-2=0

y2+2y2-1=0

y=1, y= -1

The solid lies between y=-1 and y= 1

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

V= ∫-11Aydy=∫-11π(4-4y2+y4-y8)dy

fx= 4-4y2+y4-y8 is an even function. So,

V= 2π∫01(4-4y2+y4-y8)dy

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

V=2π4y-4y33+y55-y9901

By substituting the limits of integration,

=2π 4(1)-4(1)33+(1)55-(1)99-4(0)-4(0)33+(0)55-(0)99

=2π4-43+15-19 -0

=2π 18045-6045+945-545=2π180-60+9-545=2π12445

V=24845π

Therefore, the volume of the solid obtained by rotating the region bounded by the given curves about the y axis is V=24845π

**Conclusion:**

The volume of the solid obtained by rotating the region bounded by the given curves about the y axis is V=24845π

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