#### 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= ∫0π42ππ2-xtanxdx

#### Explanation

**1) Concept:**

i. If x is the radius of the typical shell, then the circumference =2πx and the height is y.

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

V= ∫ab2πxf(x)dx

where 0≤a<b

**2) Given:**

y=tanx, y=0, x=π4; about x=π2

**3) Calculation:**

As the region is bounded by y=tanx, y=0, x=π4; about x=π2, draw the region using the given curves.

The graph shows the region. A typical cylindrical shell is formed by the rotation about the line x=π2

It has the radius =π2-x

Therefore, the circumference is 2ππ2-x and the height is tanx.

It is given that x=π4

So, a=0 and b=π4

So, the volume of the given solid is

V= ∫0π42ππ2-xfxdx

= ∫0π42ππ2-xtanxdx

**Conclusion:**

The integral for volume of solid obtained by rotating the region bounded by the given curves about the x=π2 is

V= ∫0π42ππ2-xtanxdx

#### To determine

**b) **

**To evaluate:**

The integral ofthe volume of the solid

#### Answer

≈2.25323

#### Explanation

**1) Concept:**

Use calculator of integration to evaluate integral

2) **Calculation:**

From part (a)

V= ∫0π42ππ2-xtanxdx

By using calculator of integration, (refer to exercise 21)

≈2.25323

**Conclusion:**

Therefore,

∫0π42ππ2-xtanxdx≈2.25323