MHB Area of a Cardioid - Littlehime's Question @ Yahoo Answers

[MarkFL]

Here is the question:

Help finding the area of this cardioid 10 points best?


Find the area inside the cardioid r=4+4cos(θ) for 0 less than/equal to θ less than/equal to 2pi

I have posted a link there to this thread so the OP can see my work.

 

[MarkFL]

Hello Littlehime,

The area in polar coordinates is given by:

$$A=\frac{1}{2}\int_{\alpha}^{\beta}r^2\,d\theta$$

For the given problem, we are told:

$$\alpha=0,\,\beta=2\pi,\,r=4\left(1+\cos(\theta) \right)$$

And so we have:

$$A=8\int_{0}^{2\pi} \left(1+\cos(\theta) \right)^2\,d\theta$$

Expanding the integrand, we may write:

$$A=8\int_{0}^{2\pi} 1+2\cos(\theta)+\cos^2(\theta)\,d\theta$$

Applying a double-angle identity for cosine, we obtain:

$$A=4\int_{0}^{2\pi} 4\cos(\theta)+\cos(2\theta)+3\,d\theta$$

Applying the FTOC, we get:

$$A=4\left[4\sin(\theta)+\frac{1}{2}\sin(2\theta)+3\theta \right]_0^{2\pi}=4\left(3(2\pi) \right)=24\pi$$