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 8rightrectangle approximation of the area under
from 0 to 4 and compute the approximation.

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

Figure the base and plug in.
Constants, like 1/2, can be pulled through the sigma symbol.

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.

Plug in your function,

Simplify:

Use the sum of squares rule to finish: