Question

In Exercises 21-30, sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral \((a>0, r>0)\) $$ \int_{-r}^{r} \sqrt{r^{2}-x^{2}} d x $$

Short answer

The value of the integral is \((1/2) \pi r^{2}\), which is the area of the semicircle with radius \(r\).

Step-by-step solution

Understanding the Integral

The given integral \( \int_{-r}^{r} \sqrt{r^{2}-x^{2}} dx \) looks like the formula for semicircle with radius r. Let's consider the equation of a circle in Cartesian coordinates, which is \(x^{2}+y^{2}=r^{2}\). If we solve for y, we get \(y=\sqrt{r^{2}-x^{2}}\), which is exactly the integrand.

Connecting Integral to a Geometric Shape

Recall that an integral can be interpreted as the area under the curve. Here, the integral from -r to r of \(\sqrt{r^{2}-x^{2}}\) dx is the area under the curve of the semicircle in the xy-plane from -r to r. In terms of geometry, the area of a semicircle is half the area of a full circle.

Using Geometric formula

The formula for the area of a circle is \(\pi r^{2}\). Since we're dealing with a semicircle, the area is half this, or \((1/2) \pi r^{2}\).