Why is the formula for the area of a cardioid $ \int_a^b \frac{1}{2} r^2 d \theta$ I've seen this expression in many places :$\int_a^b \frac{1}{2} r^2 d \theta$ and was wondering if someone can explain where this came from? I've noticed that it's sometimes explained in conjunction with double integrals but I'm not very familiar with double integrals (as in what it means using a picture). 

There are at least two way to look at this formula. The simplest, in my opinion, is approximating the area inside the arc into small circular arcs.
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The area inside each arc of radius $r$ and angle $\Delta\theta$ is $\frac12r^2\Delta\theta$ (this is proportional to $\pi r^2$ for the entire circle). Summing these approximations as $\Delta\theta\to0$ yields
$$
\int\frac12r^2\mathrm{d}\theta
$$

Another approach is to look at the Jacobian of the conversion from rectangular to polar coordinates.
$$
\mbox{
$\begin{align}
x&=r\cos(\theta)\\
y&=r\sin(\theta)
\end{align}
$}
\quad\text{and}\quad
\mbox{$
\begin{align}
\mathrm{d}x&=\cos(\theta)\,\mathrm{d}r-r\sin(\theta)\,\mathrm{d}\theta\\
\mathrm{d}y&=\sin(\theta)\,\mathrm{d}r+r\cos(\theta)\,\mathrm{d}\theta
\end{align}
$}
$$
That is
$$
\begin{bmatrix}\mathrm{d}x\\\mathrm{d}y\end{bmatrix}
=\begin{bmatrix}\cos(\theta)&-r\sin(\theta)\\\sin(\theta)&r\cos(\theta)\end{bmatrix}
\begin{bmatrix}\mathrm{d}r\\\mathrm{d}\theta\end{bmatrix}
$$
The determinant of the Jacobian yields
$$
\begin{align}
\mathrm{d}x\,\mathrm{d}y
&=r(\cos^2(\theta)+\sin^2(\theta))\,\mathrm{d}r\,\mathrm{d}\theta\\
&=r\,\mathrm{d}r\,\mathrm{d}\theta
\end{align}
$$
if we integrate in $r$ from the origin first, this yields the area to be
$$
\int\frac12r^2\,\mathrm{d}\theta
$$
