Question

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the curve r = √(cos θ)

Accepted Answer

First, understand the curve given in polar coordinates: $$r = \sqrt{\cos 	heta}. $$Since $$r$$ must be real, identify the domain of $$	heta$$ where $$\cos 	heta \geq 0. $$This will help determine the bounds for integration. Sketch the curve by plotting points for values of $$	heta$$ within the domain found. Note that $$r is $$non-negative and depends on $$\cos 	heta$$, which is positive in the intervals $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$ and $$[\frac{3\pi}{2}, \frac{5\pi}{2}]$$, but since $$r is $$defined as $$\sqrt{\cos 	heta}$$, focus on the principal interval where $$r is $$real and positive. Recall the formula for the area enclosed by a polar curve between angles $$\alpha$$ and $$\beta is$$ $$A = \frac{1}{2} \int_{\alpha}^{\beta} r^2$ \, d	heta. $$Substitute $$r^2$$ = \cos 	heta$$ into this formula. Set up the integral for the area as A$$ = \frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos 	heta \, d	heta$$, using the interval where r$ is real and positive. Evaluate the integral by integrating $$\cos 	heta$$ over the interval, then multiply by $$\frac{1}{2}$$. This will give the area of the region inside the curve.

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside the curve r = √(cos θ)

Verified video answer for a similar problem:

Key Concepts

Polar Coordinates and Curves

Area Calculation in Polar Coordinates

Determining the Limits of Integration