Find the area of the finite region bounded by the curve with equation x^2 - y^2 = 4 and the line x = 5.
So to find the region we've to find, int (x^2 - 4)^1/2 dx with limit 0-5. But how do I integrate this?
You will need to apply a trigonometric substitution to the integral, in particular, the substitution $\displaystyle x=2sec(\theta)$. Differentiating x, we get $\displaystyle dx=2sec(\theta)tan(\theta)d\theta$. For now, don't worry about the limits of integration. The integral will become the following:
$\displaystyle \int\sqrt{4sec^2(\theta)-4}*2sec(\theta)tan(\theta)d\theta$
Take it from there, and see how you go.