Question

Find the area inside the cardioid r=5+4cos(θ)

Answer 1

Answer:

Step-by-step explanation:

Given: $r=5+4cos{\theta}$

Area inside the cardioid is given by: A=$\int\int\limits_D {r} \, drd{\theta}$

=$\int_{0}^{2\pi}d\theta\int_{0}^{5+4cos\theta}rdr$

=$\frac{1}{2}\int_{0}^{2\pi}d\theta(r^{2})_{0}^{5+4cos\theta}$

=$\frac{1}{2}\int_{0}^{2\pi}(5+4cos\theta)^{2}d\theta$

=$\frac{1}{2}\int_{0}^{2\pi}(25+16cos^2\theta+40cos\theta)$

=$\frac{1}{2}\int_{0}^{2\pi}(25+8+8cos2\theta+40cos\theta)$

=$\frac{1}{2}\int_{0}^{2\pi}(33+8cos2\theta+40cos\theta)$

=$\frac{1}{2}(33\theta+16sin\theta+40sin\theta)_{0}^{2\pi}$

=$\frac{1}{2}((33(2\pi))+16sin(2\pi))+40sin(2\pi))$

=$33{\pi}$

Thus, the area inside the cardoid= $33{\pi}$