Mathematics, 19.03.2020 17:55 Kykebailey2356
Use a double integral in polar coordinates to find the area of the region bounded on the inside by the circle of radius 5 and on the outside by the cardioid r=5(1+cos(θ))
Answers: 1
![I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_5^{5(1+\cos \theta)} r\, dr\, d\theta\\\\I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left[ \frac{r^2}{2}\right]_5^{5(1+\cos \theta)} \, d\theta\\\\I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{25\cos \theta(\cos \theta +2)}{2}\, d\theta\\\\I=\frac{25}{2}\left[\frac{\sin 2\theta}{4}+\frac{\theta}{2}+2{\sin \theta}\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\\\\I=\frac{25}{2}\left(\frac{\pi}{2}+4\right)\\](/tpl/images/0554/2902/b831f.png)
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