Integration with Chain Rule
Definite Chain Rule Integration
This is finding the area under a curve between two specific points. It operates the exact same as evaluating normal integrals except when dealing with u substitution. When you substitute for u, if you change the bounds accordingly you do not need to substitute back in for x. See below for examples.
4.1 
4.2 Find the area under 
from x = 1 to x = 2
Substitute in terms of u for terms of x. Notice the change in interval. This is calculated by substituting in the numbers making the original interval into the equation 
. We do this because the integral is now in terms of u we need to change the interval into terms of u as well.
4.3 Find the area under 
4.4 Find the area under 
Substitute in terms of u for terms of x. Again notice the change in the interval to conform to the new variable u (not x).
