How to Do Integration by Parts
How to Solve Improper Integrals for Functions that Have Vertical Asymptotes
Integrate When the Powers of Sine, Cosine Are Even, Nonnegative

How to Do an Area Approximation Using Sigma Notation

Sigma notation comes in handy when you’re approximating the area under a curve. For example, express an 8-right-rectangle approximation of the area under

image0.png

from 0 to 4 and compute the approximation.

  1. Express the basic idea of your sum:

    image1.png

This just means that you’re adding up the areas of 8 rectangles, each of which has an area of base times height.

  1. Figure the base and plug in.

    image2.png

Constants, like 1/2, can be pulled through the sigma symbol.

  1. Add the limits of summation, and express the height as a function of the index of summation:

        

Since each rectangle has a base of 1/2, the right edge of the first rectangle will be at 1/2; the right edge of the second rectangle will be at 2/2, or 1; the right edge of the third will be at 3/2, etc. That’s what the does above. If you plug 1 then 2 then 3, etc. up to 8 into i, you get the locations of the right edges of all 8 rectangles.

  1. Plug in your function,

    image3.png
  2. Simplify:

    image4.png
  3. Use the sum of squares rule to finish:

    image5.png
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How to Do Integration by Parts More than Once
Solve Improper Integrals with One or Two Infinite Limits of Integration
Integration by Parts with the DI-agonal Method
How to Use Sigma Notation
The Sum Rule, the Constant Multiple Rule, and the Power Rule for Integration
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