Chapter 6, Problem 1GYR

To determine

Provide notes on the calculation of the volume of solids by the method of slicing with example.

Explanation of Solution

The volume of a solid of integrable cross-sectional area $A\left(x\right)$ from $x=a$ to $x=b$ is the integral of A from a to b,

$V={\displaystyle \underset{a}{\overset{b}{\int }}A\left(x\right)dx}$

Calculate the volume of a solid as follows:

• Sketch the solid and a typical cross section.
• Calculate the area of a typical cross-section.
• Calculate the limits of integration.
• Integrate the area of cross section to find the volume.

Example:

Consider the diagonal of the square runs from the parabola $y=-\sqrt{x}$ to the parabola $y=\sqrt{x}$.

Consider the solid lies between planes normal to the x axis at $x=0$ and $x=4$.

The shape of the cross section is square.

Calculate the cross sectional area $A\left(x\right)$ as follows:

$\begin{array}{c}A\left(x\right)=\frac{{\left(\text{Diagonal}\right)}^2}{2}\\ =\frac{{\left(\sqrt{x}-\left(-\sqrt{x}\right)\right)}^2}{2}\\ =2x\end{array}$

Calculate the volume of the solid using the formula:

Substitute $2x$ for $A\left(x\right)$, 0 for a, and 4 for b

$\begin{array}{c}V={\displaystyle \underset{a}{\overset{b}{\int }}A\left(x\right)dx}\\ ={\displaystyle \underset{0}{\overset{4}{\int }}2xdx}\\ =2{\left(\frac{x^2}{2}\right)}_0^4\\ =16\end{array}$

Therefore, the volume of the solid is $\underset{\_}{16}$.