Solving Integrals with Semicircles: Proving $\frac{1}{2}\pi a^2$

[tandoorichicken]

I think Mr. Gishe went trigger happy when he was assigning homework today.

$$y=\sqrt{a^2-x^2}$$ where -a<=x<=a is a semicircle with r=a
show why $$\int_{-a}^{a} \sqrt{a^2-x^2}\,dx = \frac{1}{2} \pi a^2$$

(I think that's how its written)

 

[StephenPrivitera]

What does the integral represent if the integrand is positive!? AREA!
What's the area of a half circle?

 

[himanshu121]

It is given in the que it is a semi-circle
Do you know the area of circle
if yes then u can find the area of semicircle

or if u don't know n u want to calulate the integral

substitute
x=asinz