Known cross-section volumes | Code Scholars Tutoring

Understand the idea

A solid volume is the integral of its cross-sectional area. First identify the base length from the region, then convert that length to the area of the stated shape. For a semicircle whose diameter is the base width $\displaystyle s$ , area is $\displaystyle \pi s^2/8$ . For an equilateral triangle it is $\displaystyle \sqrt3s^2/4$ . Rectangular sections require the given height relationship.

V=\int_a^b A(x)\,dx;\qquad A_{\text{square}}=s^2,\quad A_{\text{semicircle}}=\frac{\pi s^2}8

Worked example

A solid has base between $\displaystyle y=x$ and the axis on $\displaystyle 0\le x\le2$ . Sections perpendicular to the $\displaystyle x$ -axis are squares. Find its volume.

The vertical base width is

\displaystyle s=x

. A square section has area

\displaystyle x^2

, so

\displaystyle V=\int_0^2x^2\,dx=8/3

cubic units.

Try it yourself

Use semicircles with the same base widths instead of squares. Find the volume.

Show solution and reasoning

$\displaystyle A(x)=\pi x^2/8$ , so $\displaystyle V=(\pi/8)(8/3)=\pi/3$ . The width is the diameter, not the radius.