#### To determine

**(a)**

**To find:**

What is the volume of a cylindrical shell?

#### Answer

The volume of a cylindrical shell is

V=2πrh ∆r=(circumference)(height)(thickness)

#### Explanation

The volume of a cylindrical shell is

V=2πrh ∆r=(circumference)(height)(thickness)

where, r is average radius of the shell,

∆r is thickness of cylindrical shell, and

h is height of cylindrical shell.

#### To determine

**(b)**

**To explain:**

How to use cylindrical shells to find the volume of a solid of revolution.

#### Answer

We approximate the region to be revolved by rectangles, oriented so that revolution forms cylindrical shells rather than disks or washers. For a typical shell, find the circumference and height in terms of x or y and calculate.

V=limn→∞∑i=1n2π xifxi∆x = ∫ab2πxf(x)dx

V= ∫ab(circumference)(height)dx

or

V=limn→∞∑i=1n2π yifyi∆x = ∫ab2πyf(y)dy

V= ∫cdcircumferenceheightdy

#### Explanation

Consider the solid S obtained by rotating the region bounded by y=fx about y- axis, where fx≥0, y=0, x=a and x=b, where b>a≥0.

We divide the interval [a, b] into n subintervals [xi-1, xi] of equal width ∆x.

Therefore, we approximate the region to be revolved by rectangles, oriented so that revolution forms cylindrical shells rather than disks or washers. For a typical shell, find the circumference and height in terms of x or y and calculate.

V=limn→∞∑i=1n2π xifxi∆x = ∫ab2πxf(x)dx

V= ∫ab(circumference)(height)dx

or

V=limn→∞∑i=1n2π yifyi∆x = ∫ab2πyf(y)dy

V= ∫cdcircumferenceheightdy

#### To determine

**(c)**

**To explain:**

Why might you want to use shell method instead of slicing?

#### Answer

Sometimes, when disk or washer method are used, slicing produces washers or disks whose radii are difficult or sometimes impossible to find it explicitly. However, in such cases, the cylindrical shell method leads to an easier integral than slicing.

#### Explanation

Sometimes, when disk or washer method are used, slicing produces washers or disks whose radii are difficult or sometimes impossible to find it explicitly. However, in such cases, the cylindrical shell method leads to an easier integral than slicing.