Question

Find the area of the given region using any method.

Answer 1

The best method would certainly be integration :)

The area is given by the integral,

$\displaystyle\int_0^2\frac{\mathrm dx}{9-x^2}$

We can split up the integrand into partial fractions:

$\dfrac1{9-x^2}=\dfrac16\left(\dfrac1{3-x}+\dfrac1{3+x}\right)$

Then the area is

$\displaystyle\frac16\int_0^2\left(\frac1{3-x}+\frac1{3+x}\right)\,\mathrm dx$

$=\dfrac16(-\ln|3-x|+\ln|3+x|)\bigg|_0^2$

$=\dfrac16\left((\ln5-\ln1)-(\ln3-\ln3)\right)=\boxed{\dfrac{\ln5}6}$