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Home Integral Integration by Change of Variables Area of a Semicircle  
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Area of a Semicircle
We can use integration by change of variables to derive the formula for the area of a circle, A = r^{2}π, 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 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: DEFINITION 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
Therefore the semicircle has area r^{2}π/2 and the circle area r^{2}π (Figure 4.4.6). Figure 4.4.6


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