# to calculate the approximate volume of solid ( S ) by a Riemann sum. To Write: An expression for the exact volume of solid.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 6, Problem 3RCC

(a)

To determine

## To Explain: to calculate the approximate volume of solid (S) by a Riemann sum.To Write: An expression for the exact volume of solid.

Expert Solution

### Explanation of Solution

• Divide the solid in to n number of ‘slabs’ of equal width (Δx).
• Consider any one of the slab as strip (i) and area of that ith strip as A(xi*).

Find the approximate volume of the ith strip (V):

V=Widthofthestrip×Areaofthestrip

Substitute Δx for width of the strip and A(xi*) for the area of the strip.

V=i=1nA(xi*)Δx

The exact volume of the solid is the summation of the volume of all slabs.

Consider the number of “slabs”.

Find the exact volume of the solid as follows:

V=limni=1nA(xi*)Δx (1)

Rearrange Equation (1)

V=abA(x)dx

Here, the lower limit is a, upper limit is b, total area of the solid is A, and total width is x.

Thus, the exact volume of solid (S) is V=abA(x)dx_.

(b)

To determine

Expert Solution

### Explanation of Solution

Consider the revolution solid is a washer shape. The expression for the area for the washer is as follows:

π(rout2)π(rin2)

Here, rout is outer radius of washer and rin is inner radius of washer.

Consider the revolution solid is a disk shape. The expression for the area for the disk is as follows:

A=π(r)2

Here, r is radius of the disk.

Therefore, the cross-sectional area for revolution solid is defined by washer and disk method.

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