Problem 30 - Essential Calculus

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 \((a>0, r>0)\) $$ \int_{-r}^{r} \sqrt{r^{2}-x^{2}} d x $$

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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 \((1/2) \pi r^{2}\).

Key Concepts

Semicircle Area

The concept of the semicircle area arises from understanding the shape described by the given integral. A semicircle is half of a circle. To find the area of a semicircle, start with the formula for the area of a full circle, which is \(\pi r^2\). Here, \(r\) is the radius of the circle.

• To find the area of a semicircle, simply take half of the full circle's area.
• Thus, the area of a semicircle is \(\frac{1}{2} \pi r^2\).

Understanding this geometric formula is crucial when evaluating integrals that represent semicircles. With \(y = \sqrt{r^2 - x^2}\), we observe a semicircle outline from \(-r\) to \(r\) along the x-axis.

Geometric Interpretation of Integral

The geometric interpretation of a definite integral is a vital concept in understanding the link between calculus and geometry. A definite integral such as \(\int_{-r}^{r} \sqrt{r^2 - x^2} \ dx\) represents the area under the curve,which in this case forms a semicircle.

• The expression under the integral sign \(\sqrt{r^2 - x^2}\) matches the formula for a semicircle's boundary.
• The limits of the integral, \(-r\) to \(r\), correspond to the diameter of the semicircle.

This confirms that, geometrically, the integral computes the area of the semicircle in the xy-plane. Therefore, the integral is basically measuring the same area calculated using the semicircle's geometric formula.

Definite Integral Evaluation

Evaluating the definite integral \(\int_{-r}^{r} \sqrt{r^2-x^2} \ dx\) involves associating it with a known geometric area.

• The integral calculates exactly half the area of a full circle.
• Through geometric interpretation, you can directly apply the formula of a semicircle's area \(\frac{1}{2} \pi r^2\).

By recognizing the nature of the curve described by the integrand and the limits of integration, you avoid complex analytical methods.Simply relate the integral to its geometric form and utilize basic circle area calculations.Therefore, the solution results in \(\frac{1}{2} \pi r^2\), elegantly connecting calculus with basic geometry.