Okay, well I still don't really know how to integrate functions like that, and not all of them are semicircles. For instance, integrating (2-x)^6....I'm not sure how you would do that other than the backwards chain rule, and I'm not sure how you do that either...
If you draw a quarter-circle in the first quadrant, radius=2,
pick any point on the circumference not on an axis,
join the point to the origin (circle centre) and drop a perpendicular to the x-axis
then you have a right-angled triangle.
By Pythagoras' theorem
Label the acute angle at the origin
This allows you to integrate the trigonometric integral
Double your final answer.
or rename it "u".
You would have a "power" integral if it was
Now, you have a "power" integral.
It's about getting it into a form you can integrate.
The advantage of integrating the "semi-circle" function is..
since you would know the area of a semicircle,
then you will be "dead sure" your integration is incorrect if you do not get an answer of
1. You can sketch graphs (like ) by hand, without having to use a graphing calculator. If you cannot do this you are strongly advised to review pre-calculus.
2. You understand the limitations (due to resolution etc.) of graphs drawn using a graphics calculator.