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(\mathrm dx)/(9-x^2)`

We can split up the integrand into partial fractions:

`\frac1{9-x^2}=\frac16\left(\frac1{3-x}+\frac1{3+x}\right)`

Then the area is

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

`=\frac16(-\ln|3-x|+\ln|3+x|)\bigg|_0^2`

`=\frac16\left((\ln5-\ln1)-(\ln3-\ln3)\right)=\boxed{\frac{\ln5}6}`