Area between the curves

Pranav

Well-known member
Problem:

Calculate the area of region defined by the inequalities:

$$-1<xy<1$$
$$-1<x^2-y^2<1$$

Attempt:

Although I have solved the problem but I am not very satisfied with the method I used. The graph of region is symmetrical in all the four quadrants so I calculated the area in the first quadrant and multiplied by 4. I found the points of intersection and got the following integrals:

Wolfram Alpha

But I feel that given the shape of region, there exists an easier method. Has someone got any idea about a shorter method?

Here's an image depicting the region: Any help is appreciated. Thanks!

MarkFL

Staff member
I would use polar coordinates and divide the region into octants.

The curve:

$$\displaystyle x^2-y^2=1$$

becomes:

$$\displaystyle r^2=\sec(2\theta)$$

and the curve:

$$\displaystyle xy=1$$

becomes:

$$\displaystyle r^2=2\csc(2\theta)$$

we find they intersect at:

$$\displaystyle \theta=\frac{1}{2}\tan^{-1}(2)$$

and so we may express the area as:

$$\displaystyle A=8\left(\frac{1}{2}\int_0^{\frac{1}{2}\tan^{-1}(2)} \sec(2\theta)\,d\theta+\int_{\frac{1}{2}\tan^{-1}(2)}^{\frac{\pi}{4}} \csc(2\theta)\,d\theta \right)$$

You should then be able to show that:

$$\displaystyle A=2\sinh^{-1}(2)-4\ln\left(\frac{2}{1+\sqrt{5}} \right)\approx4.81211825059603$$

When you multiply the result you give from W|A by 4, you find they are equivalent.

Pranav

Well-known member
I would use polar coordinates and divide the region into octants.

The curve:

$$\displaystyle x^2-y^2=1$$

becomes:

$$\displaystyle r^2=\sec(2\theta)$$

and the curve:

$$\displaystyle xy=1$$

becomes:

$$\displaystyle r^2=2\csc(\theta)$$

we find they intersect at:

$$\displaystyle \theta=\frac{1}{2}\tan^{-1}(2)$$

and so we may express the area as:

$$\displaystyle A=8\left(\frac{1}{2}\int_0^{\frac{1}{2}\tan^{-1}(2)} \sec(2\theta)\,d\theta+\int_{\frac{1}{2}\tan^{-1}(2)}^{\frac{\pi}{4}} \csc(2\theta)\,d\theta \right)$$

You should then be able to show that:

$$\displaystyle A=2\sinh^{-1}(2)-4\ln\left(\frac{1}{1+\sqrt{5}} \right)\approx4.81211825059603$$

When you multiply the result you give from W|A by 4, you find they are equivalent.
That's beautiful MarkFL! Thanks a bunch.   You seem to have made a very small error in the end. I get $4\ln\left(\frac{2}{1+\sqrt{5}}\right)$ instead of $4\ln\left(\frac{1}{1+\sqrt{5}}\right)$, rest everything is great. Thanks once again.

MarkFL

That's beautiful MarkFL! Thanks a bunch.   You seem to have made a very small error in the end. I get $4\ln\left(\frac{2}{1+\sqrt{5}}\right)$ instead of $4\ln\left(\frac{1}{1+\sqrt{5}}\right)$, rest everything is great. Thanks once again.
Yes, that was a typo, when I entered the expression into the calculator, I used the correct number. I will fix my post. I also had a typo when I stated the polar equivalent of $xy=1$, but I used the correct form in the integrand. I have now fixed that as well.