Riemann+Sums+and+Trapezoidal+Approximations

** Riemann sums and trapezoidal approximations are simple ways of approximating the area under a curve. To do a Riemann Sum, the function is divided into n subintervals. Each subinterval contains a sample point where the corresponding function value forms the altitude of the rectangle. The area of each rectangle then becomes the width of the rectangle times the altitude (the corresponding function value of the sample point). Then, you just add up all the rectangles. Riemann sums can be left, right, midpoint, upper or lower. Most are self explanatory: **Left Riemann sum:** To find this Riemann Sum, in each subinterval your sample points are the points furthest to the left. This is an underestimate because as you can see in each interval a portion under the graph is excluded form your calculations. **Right Riemann sum:** The same is done for the right Riemann sum but this time the sample points are taken from the points furthest to the right of the increment. This is an overestimate because the rectangles include the area under the curve as well as some above it. **Midpoint Riemann sum:** Again the same process, but this time the sample point is taken at the midpoint of each subinterval. **Upper and Lower Riemann sum:** Same process taking the highest point in each subinterval, and a lower Riemann Sum taking the lowest (seen on the right).
 * Riemann Sums and Trapezoidal Approximations



These approximations are the same basic idea as the problems we've done so far except instead of creating rectangle and adding them up, it's trapezoids. In each subinterval, the endpoints on each side (on the actual function) are connected forming a trapezoid. The area of these trapezoids are calculated by multiplying one half times the sum of the function values of each endpoint (the sides of the trapezoid) times the width of the subinterval. It's easier to visualize:
 * Trapezoidal Approximations:**



This is an overestimate if the graph is concave up and an underestimate if it's concave down. **

It's pretty long and the guy is weird, but it covers mostly everything: ** media type="youtube" key="YHYT3HlL1uA" height="385" width="480"

__Sources__ Pictures from: [|Wikipedia] [|HMC Mathematics Online Tutorial]

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