The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

Area of a Semicircle

Example 7: Area Under A Semicircle

We can use integration by change of variables to derive the formula for the area of a circle, A = r2π, where r is the radius. It is easier to work with a semicircle because the semicircle of radius r is just the region under the curve

04_integration-277.gif r ≤ x ≤ r.

To start with we need to give a rigorous definition of π. By definition, π is the area of a unit circle. Thus π is twice the area of the unit semicircle, which means:



The area of a semicircle of radius r is the definite integral


To evaluate this integral we let x = ru. Then dx = r du. When x = ±r,u = ±1. Thus

04_integration-274.gif 04_integration-272.gif

Therefore the semicircle has area r2π/2 and the circle area r2π (Figure 4.4.6).


Figure 4.4.6

Last Update: 2010-11-26