Find the area between two polar curves

Question

I need to find the area between two polar curves, $$r = \frac1{\sqrt{2}}$$ $$r = \sqrt{\cos(θ)}$$

I've found the intersections to be at $\fracπ3$ and $\frac{5π}3$, and I've set up the equation to find the area as

$$\int\limits_{\fracπ3}^{\frac{5π}3} \sqrt{\cos(θ)}^2 - \frac1{\sqrt{2}}^2 \, \mathrm dθ,$$

but whenever I plug it into a calculator, it comes up as undefined, so it can't possibly be correct. Could you help me with this?

Answer 1

The area show in green

is given by the following integral, where $r(\theta)=\sqrt{\cos\theta}$ $$ \int_{\pi/3}^{\pi/2}d\theta\int_0^{r(\theta)}rdr=\frac{1}{2}\int_{\pi/3}^{\pi/2}\cos\theta d\theta=\frac{1}{2}\left(1-\frac{\sqrt{3}}{2}\right) $$

The rest should be pretty easy.

Answer 2

If you plot both functions, and restrict the domain of r = $\sqrt{cos \theta}$ you see that from $\frac{\pi}{3}$ to $\frac{\pi}{2}$ you get half of the inner function. So integrate $1/2 - \cos \theta$ between those limits and double your answer to get the full area. I get $\frac{\pi}{6} - 2 + \sqrt{3}$.

Answer 3

Try running it as two separate integrals, each with their own integrand and dtheta. Evaluate each integrand and then subtract them from each other. However, make sure you have the problem written down exactly the same - the error is not with the formatting but that this equation only has one imaginary answer (~ -2.4616 + 2.3962i). If your problem does not ask for an imaginary anwer for answer for this specific equation, its not defined.