#### 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= 256π5

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

#### Explanation

**1) Concept:**

i. If the cross section is a disc and the radius of the disc is in terms of x or y then the area A=π radius2

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

V= ∫abA(y)dy

**2) Given:**

The region is bounded by 2x=y2, x=0, y=4 about the y - axis

**3) Calculation:**

The region bounded by 2x=y2, x=0, y=4 and rotated 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 a disk with radius y22.

So, its cross sectional area becomes

Ay=πy222=14πy4

The solid lies between y=0 and y= 4

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

V= ∫04Aydy=∫0414πy4dy

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

V=π4y5504

By substituting the limits of integration,

=π4455-0

=π445

V=256π5

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

V=256π5

**Conclusion:**

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

V=256π5

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