8.8 Volumes with Cross Sections: Triangles and Semicircles

Setting up the cross section

When a solid has a base region and cross sections perpendicular to an axis, the volume is found by integrating the area of each slice along that axis: V = ∫ A(x) dx. First find the length of the slice using the boundary curves (this length becomes a side, diameter, or leg depending on the shape), then plug it into the correct area formula.

Triangle cross sections

If the slice length s (from the base region) is a side of an equilateral triangle, area = (√3/4)s². If it's the leg of an isosceles right triangle (leg on the base), area = (1/2)s². If s is the hypotenuse of an isosceles right triangle, area = (1/4)s². Always identify which part of the triangle the base region represents before choosing the formula.

Semicircle cross sections

If the slice length s is the diameter of a semicircle, the radius is s/2, so area = (1/2)π(s/2)² = πs²/8. Set up the integral V = ∫ πs²/8 dx (or dy) between the bounds, where s = s(x) is the distance between the top and bottom curves of the base region.

Key terms

Volume with cross sections formula

V = ∫ A(x) dx (or dy), where A(x) is the area of the cross section at each x.

Area of equilateral triangle, side s

A = (√3/4)s²

Area of isosceles right triangle, leg s (on base)

A = (1/2)s²

Area of isosceles right triangle, hypotenuse s (on base)

A = (1/4)s²

Area of semicircle with diameter s

A = πs²/8 (since radius = s/2, area = (1/2)π(s/2)²)

Try an AP-style question

What is the area of an equilateral triangle with side s?

1. (√3/2)s²
2. (√3/4)s²
3. (1/2)s²
4. s²