Question

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))

Answer 1

Answer:

$\boxed{I=\frac{25}{2}\left(\frac{\pi}{2}+4\right)}$

Step-by-step explanation:

We have:

$r=5\\\\r=5(1+\cos \theta)\\\\\implies 5=5(1+\cos \theta)\\\\\cos \theta =0\\\\\theta=\pm \frac{\pi}{2}$

We get the limits of integration:

$D=\{(r, \theta):\, 5\leq r\leq 5(1+\cos \theta), \, -\frac{\pi}{2}\leq \theta \leq \frac{\pi}{2}\}$

We use the polar coordinates and we calculate  a double integral:

$I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_5^{5(1+\cos \theta)} r\, dr\, d\theta\\\\I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left[ \frac{r^2}{2}\right]_5^{5(1+\cos \theta)} \, d\theta\\\\I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{25\cos \theta(\cos \theta +2)}{2}\, d\theta\\\\I=\frac{25}{2}\left[\frac{\sin 2\theta}{4}+\frac{\theta}{2}+2{\sin \theta}\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\\\\I=\frac{25}{2}\left(\frac{\pi}{2}+4\right)$

So,  we get:

$\boxed{I=\frac{25}{2}\left(\frac{\pi}{2}+4\right)}$