#### 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=∫2π3π2πxsinxdx

#### Explanation

**1) Concept:**

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

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= ∫ab2πx f(x)dx

Where, 0≤a<b

**2) Given:**

y=sinx, y=0, x=2π, x=3π, rotate about y-axis

**3) Calculation:**

As the region is bounded by y=sinx, y=0, x=2π, x=3π, and rotated about y-axis, draw the region using the given curves.

The graph shows the region. A typical cylindrical shell formed by rotation about the line y-axis has the radius =x.

Therefore, the circumference is 2πx and the height is sinx

It is given that x=2π, x=3π

So, a=2π and b=3π

So, the volume of the given solid is

V= ∫2π3π2πxsinx dx

**Conclusion:**

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

V=∫2π3π2πxsinx dx

#### To determine

**b) **

**To evaluate:**

The integral ofthe volume of the solid

#### Answer

98.69604

#### Explanation

**1) Concept:**

Use calculator of integration to evaluate integral

2) **Calculation:**

From part (a)

V=∫2π3π2πxsinxdx

Evaluate the integral by using a calculator Ti-84 plus.

Use the below steps:

1) Click on “Math” option.

2) After that under “Math” search for the option “fnInt(“

3) Write the lower limit, upper limit and function using the proper brackets and required variable.

4) Press enter key; it will display the final answer.

=98.69604

**Conclusion:**

Therefore,

∫2π3π2πxsinxdx=98.69604