Proving integral of sqrt(a^2 - x^2) using trig sub

[jman2807]

I am having trouble proving the integration of

sqrt(a^2 - x^2)

I know i need to use trig subsitution but i am still having trouble getting started. Any help would be appreciated. Thanks in advance.

 

[soroban]

Re: Proving interval using trig sub

Hello, jman2807!

\(\displaystyle \L\int\)\(\displaystyle \sqrt{a^2\,-\,x^2}\,dx\)

Let \(\displaystyle x \,=\,a\cdot\sin\theta\;\;\Rightarrow\;\;dx\,=\,a\cdot\cos\theta\,d\theta\)

The radical is: \(\displaystyle \:\sqrt{a^2\,-\,x^2} \:=\:\sqrt{a^2\,-\,a^2\cdot\sin^2\theta} \:=\:\sqrt{a^2(1\,-\,\sin^2\theta)} \:=\;\sqrt{a^2\cdot\cos^2\theta} \:=\:a\cdot\cos\theta\)

Substitute: \(\displaystyle \L\int\)\(\displaystyle (a\cdot\cos\theta)(a\cdot\cos\theta\,d\theta) \:=\:a^2\L\int\)\(\displaystyle \cos^2\theta\,d\theta\)

Can you finish it now?