Problem 58 - Calculus: Early Transcendental Functions

Question

Use a geometric formula to compute the integral. $$\int_{-3}^{0} \sqrt{9-x^{2}} d x$$

Step-by-Step Solution

Answer

By replacing the integral with the semicircle area formula, the area, which represents the value of the integral, is \(\frac{1}{2} \pi (3)^2 = \frac{9}{2} \pi\).

Step 1: Identifying the Geometric Figure

The given integrand, \(\sqrt{9 - x^{2}}\), represents a semicircle. So, instead of thinking of it as a traditional integral, it could be approached as a geometric problem. The definite integral will be equal to the area of this semicircle.

Step 2: Calculating the Area of a Semicircle

The formula to find the area of a semicircle is \(\frac{1}{2} \pi r^2\), where r represents the radius of the circle. In this case, the radius of the semicircle is 3 (since the integrand yields a circle with radius 3 when the square root is not applied). Substituting the radius into the formula, we can calculate the area of the semicircle.

Step 3: Evaluating the Area

Substitute the radius value into the semicircle area formula: \(\frac{1}{2} \pi (3)^2\). Simplify this expression to obtain the area.

Key Concepts

Semicircle

The concept of a semicircle is crucial when discussing geometric integration. A semicircle is exactly half of a full circle, meaning its area is half of the area of a complete circle. In problems involving functions like \( \sqrt{9 - x^2} \), the shape of the graph is a semicircle. This is because the equation \( x^2 + y^2 = r^2 \) describes a circle, and when solved for \( y \) as \( y = \sqrt{r^2 - x^2} \), it represents the upper half—forming a semicircle. Recognizing the connection between the integrand and a geometric shape simplifies the approach to solving integrals by viewing them as area calculations.

Area Calculation

Calculating the area of geometric shapes, such as a semicircle, can often replace more complex integral evaluation. The area of a semicircle can be determined using the formula \( \frac{1}{2} \pi r^2 \). This formula arises because the area of a full circle is \( \pi r^2 \). Since a semicircle is half of a circle, you simply take half of that area. By substituting the radius into this formula, you directly compute the area without needing further calculations. This approach is particularly useful in geometric integration when the function represents part of a simple geometric figure.

Definite Integrals

Definite integrals can often calculate the area under a curve between two specific points. In our case, integrating \( \sqrt{9 - x^2} \) from \(-3\) to \(0\) calculates the area under a semicircular arc on a Cartesian plane. While typically the indefinite integral of a function would provide a general solution, definite integrals include limits to provide a specific area. By understanding the geometric shape, recognizing the semicircle allows you to apply geometric formulas directly instead, simplifying the process significantly and avoiding unnecessary complexity often associated with integral calculus.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its perimeter. Understanding the role of the radius helps in solving problems involving circles and semicircles. Specifically, for the function \( \sqrt{9 - x^2} \), recognizing that the equation originates from \( x^2 + y^2 = 9 \) implies a circle with a radius of \(3\). This understanding is crucial because the radius directly substitutes into the area formula \( \frac{1}{2} \pi r^2 \) for semicircle calculations. Comprehending how the radius fits into these equations is essential for solving geometric integrals effectively.