59. Area of a segment of a circle
Use two approaches to show that the area of a cap (or segment) of a circle of radius r subtended by an angle θ (see figure) is given by:
A_seg = (1/2) r² (θ - sin θ)
b. Find the area using calculus.

Verified step by step guidance

1. Step 1: Understand the problem setup. We want to find the area of the segment (or cap) of a circle with radius \(r\) subtended by a central angle \(\theta\). The segment is the region bounded by the chord and the arc corresponding to \(\theta\).
2. Step 2: Express the circle in Cartesian coordinates. Place the circle centered at the origin with equation \(x^2 + y^2 = r^2\). The segment lies above the chord, so we consider the upper semicircle: \(y = \sqrt{r^2 - x^2}\).
3. Step 3: Determine the limits of integration. The chord endpoints correspond to the angle \(\pm \frac{\theta}{2}\) from the positive x-axis. The x-coordinates of these points are \(x = r \cos \frac{\theta}{2}\) and \(x = r\) (assuming the segment is on the right side). We integrate from \(x = r \cos \frac{\theta}{2}\) to \(x = r\).
4. Step 4: Set up the integral for the area of the segment. The area under the curve (arc) from \(x = r \cos \frac{\theta}{2}\) to \(x = r\) is \(\int_{r \cos \frac{\theta}{2}}^{r} \sqrt{r^2 - x^2} \, dx\). The area of the triangle formed by the chord and the radius lines is \(\frac{1}{2} r^2 \sin \theta\). The segment area is the difference between the sector area and the triangle area.
5. Step 5: Use calculus to evaluate the integral and subtract the triangle area. The integral \(\int \sqrt{r^2 - x^2} \, dx\) can be solved using a trigonometric substitution, and after evaluating the definite integral and subtracting the triangle area, you will arrive at the formula for the segment area: \(A_{seg} = \frac{1}{2} r^2 (\theta - \sin \theta)\).

Key Concepts

Area of a Sector of a Circle

The area of a sector of a circle with radius r subtended by an angle θ (in radians) is given by (1/2)r²θ. This formula represents the portion of the circle's area enclosed by two radii and the arc between them, serving as a foundational step in finding the segment area.

Area of a Triangle Using Trigonometry

The triangle formed by the two radii and the chord can be calculated using the formula (1/2)r²sinθ, where θ is the angle between the radii. This area is subtracted from the sector area to find the segment area, linking trigonometric functions with geometric areas.

Definite Integration in Polar Coordinates

Using calculus, the area of the segment can be found by integrating the function describing the circle's boundary in polar coordinates. The integral from 0 to θ of (1/2)r² dθ minus the triangle area yields the segment area, demonstrating the application of definite integrals to find areas bounded by curves.