Question

59. Area of a segment of a circleUse 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.

Accepted Answer

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 $$	heta. $$The segment is the region bounded by the chord and the arc corresponding to $$	heta. $$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: $$y = \sqrt{r^2$ - x^2$}. $$Step 3: Determine the limits of integration. The chord endpoints correspond to the angle $$\pm \frac{	heta}{2}$$ from the positive x-axis. The x-coordinates of these points are $$x = r \cos \frac{	heta}{2}$$ and $$x = r ($$assuming the segment is on the right side). We integrate from $$x = r \cos \frac{	heta}{2} to$$ $$x = r. $$Step 4: Set up the integral for the area of the segment. The area under the curve (arc) from $$x = r \cos \frac{	heta}{2} to$$ $$x = r is$$ $$\int_{r \cos \frac{	heta}{2}}^{r} \sqrt{r^2$ - x^2$} \, dx. $$The area of the triangle formed by the chord and the radius lines is $$\frac{1}{2} r^2$ \sin 	heta. $$The segment area is the difference between the sector area and the triangle area. 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: $$A_{seg} = \frac{1}{2} r^2$ (	heta - \sin 	heta).$$

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.

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Concetti chiave

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Area of a Triangle Using Trigonometry

Definite Integration in Polar Coordinates