8.8 Volumes with Cross Sections: Triangles and Semicircles

If a solid has a known base region and each cross section perpendicular to an axis is a triangle or a semicircle, then the volume is found by integrating the cross-sectional area. The setup always starts the same way: identify the length of the cross-sectional base from the region, write the area formula for the given shape, and integrate that area across the interval. For triangles, the area depends on the exact type of triangle given, such as equilateral or isosceles right. For semicircles, the area is based on the radius, which is usually half of the given diameter.

The main skill is converting a geometric description into a usable area formula. If the cross section is an equilateral triangle with side length s, then ${A=\frac{\sqrt{3}}{4}s^2}$. If the cross section is a semicircle with diameter d, then the radius is ${\frac{d}{2}}$, so the area is ${A=\frac{1}{2}\pi\left(\frac{d}{2}\right)^2=\frac{\pi d^2}{8}}$. After that, volume is just ${V=\int A(x)\,dx}$ or ${V=\int A(y)\,dy}$. This is still a cross-sections problem, but the shape formulas are slightly more involved than for squares and rectangles.