#### 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∫-335-y4-y2+7dy

#### 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= ∫ab2πx f(x)dx

where 0≤a<b

**2) Given:**

x2-y2=7, x=4 rotates about y=5

**3) Calculation:**

As the region is bounded by x2-y2=7, x=4 and rotates about y=5, draw the region using the given curves.

The graph shows the region. A typical cylindrical is shell formed by the rotation about the line y=5

It has the radius =5-y

Now, find the height.

x2-y2=7

Solve for x from this.

x=7+y2

Now, subtract x=7+y2 from x=4 to get height.

Therefore, height 4-7+y2

Therefore, the circumference is 2π5-y and the height is 4-y2+7

Now, to find intersection points, equate both equations.

y2+7=4

y2+7=16

y2=9

y=-3, 3

So, a=-3 and b=3

So, the volume of the given solid is

V=∫-332π5-yfxdy

=2π∫-335-y4-y2+7dy

**Conclusion:**

The integral for the volume of the solid obtained by rotating the region bounded by the given curves about the y=5 is

V=2π∫-335-y4-y2+7dy

#### To determine

**b) **

**To evaluate:**

The integral of thevolume of the solid.

#### Answer

≈163.02712

#### Explanation

**1) Concept:**

Use calculator to evaluate integral

**2) Calculation:**

From part (a),

V=2π∫-335-y4-y2+7dy

By using calculator,

≈163.02712

**Conclusion:**

Therefore,

2π∫-335-y4-y2+7dy≈163.02712