8.8 Volumes: Triangle & Semicircle Cross Sections

Same integral, new area

Still V = ∫ A(x) dx, but A depends on the shape. Equilateral triangle (side s): A = (√3/4)s². Right isosceles triangle (legs s): A = (1/2)s².

Semicircle

Semicircle with diameter s: A = (π/8)s². Plug in the correct area formula, then integrate over [a, b].

Key terms

Same method

V = ∫ A(x) dx with a shape-specific area.

Equilateral triangle

A = (√3/4)s².

Right isosceles triangle

A = (1/2)s² (legs s).

Semicircle

A = (π/8)s² (diameter s).

Key

Use the correct area formula for the shape.

Try an AP-style question

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

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