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Integration

Riemann Sums

A Riemann sum is a method for approximating the area under a curve (a less accurate integral). The two main types are rectangular and trapezoidal. Rectangular entails you dividing up and area into rectangles and then adding all the rectangles up to get an area. Trapezoidal is the same but instead of dividing the area along the curve into rectangles you make them into trapezoids.

In finding Riemann sums, follow these steps.

1. Divide the distance x is covering by the number of intervals. This is your Δx, the width of the rectangle (rectangular), or height of the trapezoid (trapezoidal).
2. If you are solving for rectangular, you will be told left, right or middle.
   1. If left rectangular, calculate the area of each rectangle using the left most y value (f(x) with the smallest x of the interval) as the height of the rectangle.
   2. If right rectangular, calculate the area of each rectangle using the right most y value (f(x) with the largest x of the interval) as the height of the rectangle.
   3. If middle rectangular, calculate the area of each rectangular using the middle y value (f(x) with the x that is the average of each end of the interval) as the height of the rectangle.
3. If you are solving for trapezoidal use the y value at the left of interval as base 1 and the  y value at the right end of the interval as base 2 ().

For example:

3.1   Calculate a left rectangular Riemann sum of the function  from 1 to 3 using 5 rectangles.

3.2   Calculate a trapezoidal Riemann sum of the function f(x) = x³ from 0 to 3 using six trapezoids.

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