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.

Explanation of Solution

• Divide the solid in to n number of ‘slabs’ of equal width $\left(\Delta x\right)$.
• Consider any one of the slab as strip (i) and area of that $i^{th}$ strip as $A\left(x_i^{*}\right)$.

Find the approximate volume of the i^th strip (V):

$V=\text{Width}\text{of}\text{the}\text{strip}\times \text{Area}\text{of}\text{the}\text{strip}$

Substitute $\Delta x$ for width of the strip and $A\left(x_i^{*}\right)$ for the area of the strip.

$V={\displaystyle \sum _{i=1}^{n}A\left(x_i^{*}\right)\Delta x}$

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

Consider the $\infty$ number of “slabs”.

Find the exact volume of the solid as follows:

$V=\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}A\left(x_i^{*}\right)\Delta x}$ (1)

Rearrange Equation (1)

$V={\displaystyle {\int }_a^{b}A\left(x\right)dx}$

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 $\underset{\_}{V={\displaystyle {\int }_a^{b}A\left(x\right)dx}}$.

(b)

To determine

To Find: The cross-sectional areas if S is a solid of revolution, (S)

Explanation of Solution

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

$\pi \left(r_{\text{out}}^2\right)-\pi \left(r_{\text{in}}^2\right)$

Here, $r_{\text{out}}$ is outer radius of washer and $r_{\text{in}}$ is inner radius of washer.

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

$A=\pi {\left(r\right)}^2$

Here, r is radius of the disk.

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