Question

45–60. Areas of regions Find the area of the following regions.The region inside one leaf of the rose r = cos 5θ

Accepted Answer

Recognize that the given curve is a rose curve defined by the polar equation $$r = \cos(5	heta). $$This curve has 5 petals because the coefficient of $$	heta$$ inside the cosine is 5. Recall that the area $$A$$ enclosed by one petal of a rose curve $$r = \cos(k	heta) ($$where $$k is an $$integer) can be found using the integral formula for polar areas:

$$A = \frac{1}{2} \int_{\alpha}^{\beta} r^2$ \, d	heta$$ Determine the limits of integration $$\alpha$$ and $$\beta$$ that correspond to one leaf (one petal) of the rose. Since the rose has 5 petals, one petal corresponds to an interval of $$\frac{2\pi}{5}$$ radians. You can find the exact interval by solving $$r = 0 to $$find where the petal starts and ends, or use symmetry and set $$	heta$$ from $$0 to$$ $$\frac{\pi}{5}. $$Set up the integral for the area of one petal:

$$A = \frac{1}{2} \int_{0}^{\frac{\pi}{5}} \left( \cos(5	heta) \$$right)^2$$ \, d	heta$$ Use the trigonometric identity $$\cos^2$ x = \frac{1 + \cos(2x)}{2} to $$simplify the integrand before integrating. Then, integrate with respect to $$	heta$$ over the chosen limits to find the area of one petal.

45–60. Areas of regions Find the area of the following regions.

The region inside one leaf of the rose r = cos 5θ

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Key Concepts

Polar Coordinates and Graphs

Area in Polar Coordinates

Determining Limits of Integration for One Petal