#### To determine

**a) **

**To find:**

The integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.

#### Answer

V=2π∫0π4-ysinydy

#### Explanation

**1) Concept:**

i. If y is the radius of the typical shell, then the circumference =2πy and the height is x=f(y)

ii. By the shell method, the volume of the solid by rotating the region under the curve x=f(y) about x- axis from a to b is

V= ∫ab2πxf(y)dy

where, 0≤a<b

**2) Given:**

x=siny, 0≤y≤π, x=0; about y=4

**3) Calculation:**

As the region is bounded by,

x=siny, 0≤y≤π, x=0; about y=4

Draw the region using the given curves.

The graph shows the region and the cylindrical shell formed by the rotation about the line y=4

It has the radius =4-y

Therefore, the circumference is 2π4-y and the height is siny

It is given that

0≤y≤π

So, a=0 and b=π

So, the volume of the given solid is

V= ∫0π2π 4-yfx dy

= 2π∫0π4-ysinydy

**Conclusion:**

The integral for the volume of the solid obtained by rotating the region bounded by the given curves about the y=4 is

V=2π∫0π4-ysinydy

#### To determine

**b) **

**To evaluate:**

The integral ofthe volume of the solid

#### Answer

≈36.57476

#### Explanation

**1) Concept:**

Use calculator of integration to evaluate integral

**2) Calculation:**

From part (a)

V=2π∫0π4-ysinydy

By using calculator (refer to exercise 21),

≈36.57476

**Conclusion:**

Therefore,

2π∫0π4-ysiny dy≈36.57476