Question

Find the area as an integral in polar coordinates. r=4sec(theta-pi/4)

Answer 1

Are there bounds for \(\theta\)?

Answer 2

No

Answer 3

The reason I ask is because \(r=\sec\theta\) is the equation of a line, since
\[r=\sec\theta~~\iff~~r\cos\theta=1~~\iff~~x=1\]
which means you're going to need some restriction for the domain, otherwise the area is infinite.

Answer 4

Changing to rectangular coordinates (so that we can see what the graph looks like more easily) you would have
\[\begin{align*}r&=4\sec\left(\theta-\frac{\pi}{4}\right)\\
r\cos\left(\theta-\frac{\pi}{4}\right)&=4\\
r\bigg[\cos\theta\cos\frac{\pi}{4}+\sin\theta\sin\frac{\pi}{4}\bigg]&=4\\
\frac{1}{\sqrt2}r\bigg[\cos\theta+\sin\theta\bigg]&=4\\
r\cos\theta+r\sin\theta&=4\sqrt2\\
x+y&=4\sqrt2\end{align*}\]
which is a line that looks like this:

I think the shaded region is the area you want.