Question

Evaluate the integral below by interpreting it in terms of a partial circle area. In other words, draw a picture of the region the integral represents, and find the area using high school geometry.

-9 to 9 sqrt(81-x^2)dx

Answer 1

The first thing you want to do is make a u substitution.
Let u=2x+1
then du=2dx or dx=(1/2)du
Re-evaluate your limits plugging in zero you get 1 and plugging in 1 your get 3. So you have:

36/2 (lim 1 to 3)∫ du/u^3

18∫u^3du 18u^4/4 (18(2x+1)^4/4) 18(2(1)+1)^4/4-18(2(0)+1)^4/4 364.5-4.5