Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. $$ \int_{-3}^{3} \sqrt{9-x^{2}} d x $$

Step-by-Step Solution

Answer

The area represented by the integral is \(4.5π\).

Step 1: Sketch the function

Start by plotting the function \(\sqrt{9-x^{2}}\). This represents a semi-circle with radius 3 (the square root of 9), centered at the origin and lying above the x-axis. Similarly, \(-\sqrt{9-x^{2}}\) would represent a semi-circle below the x-axis. Together they form a full circle of radius 3.

Step 2: Identify the region represented by the integral

Next, identify that the definite integral \(\int_{-3}^{3} \sqrt{9-x^{2}} d x\) represents the area of the region under the curve \(\sqrt{9-x^{2}}\) from x=-3 to x=3. In the context of this function, this region is a semi-circle of radius 3.

Step 3: Use a geometric formula to evaluate the integral

With the region being a semi-circle, its area can be calculated with the formula for the area of a semi-circle: \(0.5 * π*radius^{2}\). So substituting 3 for the radius, we get \(0.5 * π * 3^{2} = 4.5π\).

Key Concepts

Area Under a Curve

Understanding the area under a curve is a fundamental concept in calculus, particularly when dealing with definite integrals. A definite integral, such as \( \int_{a}^{b} f(x) \, dx \), essentially provides the total area under the curve of \( f(x) \) from \( x = a \) to \( x = b \) bounded by the x-axis. This concept is crucial in various scientific and engineering fields, as it represents quantities such as the distance traveled by an object under a velocity-time graph, or the work done under a force-displacement graph.

In our exercise, the definite integral \( \int_{-3}^{3} \sqrt{9-x^{2}} \, dx \) defines the area under the curve of \( \sqrt{9-x^{2}} \), a semi-circle, between the points \( -3 \) and \( 3 \). This is a geometrically shaped curve, making the process of finding the area less abstract as it aligns with known geometric formulas. Nonetheless, the integral emphasizes the continuous nature of area computation across the curve, even for more complex functions where geometry alone might not suffice.

Geometric Interpretation of Integrals

The geometric interpretation of integrals allows students to visualize calculus concepts and understand the practical applications of integration. An integral can represent physical quantities such as area, volume, or mass, depending on the problem at hand and the function being integrated. When the function is positive over the interval, the integral corresponds to the actual geometric area bounded by the curve and the x-axis.

This visual perspective helps in comprehending complex processes. For example, integrals are not limited to finding areas under simple curves like straight lines or parabolas but can also extend to shapes like circles, ellipses, or irregular curves. When the curve resembles a known shape, as with the semi-circle in our example, the geometric formula can be directly applied after determining the bounds of the integral and the defining characteristics of the shape, such as radius for a circle.

Semi-Circle Area Calculation

Calculating the area of a semi-circle is a straightforward application of geometry. The area of a circle is given by the formula \( \pi r^{2} \), where \( r \) is the radius. Since a semi-circle is half of a full circle, its area is half the area of a full circle, hence the formula \( 0.5 \pi r^{2} \).

In our problem, the definite integral represents the area of a semi-circle with a radius of 3 since the function \( \sqrt{9-x^{2}} \) describes the upper half of a circle centered at the origin with a radius \( \sqrt{9} \) or 3 units. So, to find the area, we simply plug the radius into our semi-circle area formula, yielding \( 0.5 \cdot \pi \cdot 3^{2} \), which simplifies to \( 4.5\pi \). This melding of geometry and calculus not only simplifies the calculation in such cases but also strengthens the conceptual link between the disciplines, aiding in the comprehension and retention of both geometric and analytical methods.