I need to find the area between these two spirals given in the polar coordinates: $$r = e^{5 \theta}$$ $$r = e^{10 \theta}$$ $$0 \le \theta \le 3\pi$$ This seems to be quite simple, yet this problem is marked as "hard" and so I am not sure of the solution.
I think that it is enough to find the area between these two exp curves and then 'convert' this area into the polar coordinates.
The Jacobian is $r$.
The are will be
$$\int_{\theta = 0}^{\theta = 3 \pi} \int_{r = e^{5 \theta}}^{r = e^{10 \theta} } r drd\theta$$
Is my method a correct way to solve this? If not, please, tell me where the mistake is before posting your own, different solution.
It depends on what is meant by "the area between the two spirals". The way you're interpreting it, you're closing off the area with a border between $(3\pi,\mathrm e^{5\cdot3\pi})$ and $(3\pi,\mathrm e^{10\cdot3\pi})$ and calculating the area of the resulting closed shape. But in a sense, the area between $r=\mathrm e^{10\theta}$ for $0\le\theta\le\pi$ and $r=\mathrm e^{5\theta}$ for $2\pi\le\theta\le3\pi$ is also "between the two spirals". So I think what's "hard" here is the interpretation of the question. Under your interpretation, your solution seems correct.