This is one of the most important lessons you will teach to your Algebra 2 students because it serves as an intro to an enormous lesson in PreCalculus / Calulus. This is the first time they get to think outside the box of their typical area formulas. Now they are given restraints, conditions, and boundaries they must adhere to that really require them to think critically.
Depending on the skill level of your students, you should be able to get them at the end of the lesson to have questions regarding shapes and conditions that would require integrations (Area under NonNormal Curves). If you taught the lesson well they should have questions at the end asking "what if!"
Then you get to explain to them they will later be learning about integration which will help them solve the problems they are asking about.
This lesson is one that my students love every year because they really get to think. It's not boring! These are like puzzles. If you have them work together in small groups to find the area under the curve they will come up with very high level questions when they get stuck.
Class Activity
 A great way to do this is to put them in groups of 2 or 3 and place rectangles of the pieces of the cut out shaded area on the desk (Can be made of paper, poster board is better).
 Then have a piece of poster board on the desk (the coordinate plane can be predrawn for less advanced students).
 Have them try to find the Area of each Rectangle by placing each rectangle under the curve you gave them. They will get the side lengths from where they are placed on the coordinate plane. Have them place the shape on the coordinate plane given restraints that you give them. So that there is only one correct place to put each.
 Then have them estimate the area of the shape using the guide lines you gave them. Obviously you will want them to start basic but you can make the shapes however complicated you want as long as they don't require integration.
TIPS:
 Use common curves
 Make the spacing of all of the rectangles for all groups and all curves the width of a yard stick so they are consistent
 Set small boundaries at first (ie: 0 to 10 where the width of one rectangles is 1 unit)
 Once they have an estimate for the curve with boundaries ask them what would happen if you went 0 to 100 or 0 to 1000? Leading them to 0 to infinity!
You can then have them switch shapes with another group and do it again. This keeps them focused on one problem at a time. Students today have short attention spans. Control this by only having them see one problem at a time so they do not think they have to do a bunch more after the first and give up.
Sigma notation is completely new outside the box thinking for them. Ease into it as much as possible. You don't want to lose them before you get into the good stuff!
Approximating the Area Under the Curve Using Sigma Notation
Real World Uses for Area Under a Curve
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