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We have 2 ways that we commonly approximate area. Riemann (rectangle) sums are the first way and Trapezoidal approximations are the second way.
Riemann sums can be calculated several different ways. Left, Right, Midpoint, Over-approximation, and Under-approximation are the most common 5 ways to determine the height of each of the rectangles. See example for more.
Trapezoidal sums are calculated by simply finding the areas of all the trapezoids and adding them together. Trap Area = Average of the Bases * Height. If you are having trouble determining the bases, redraw the trapezoid on your paper. The Parallel sides are the bases.
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