A Riemann sum is a method for approximating the total area underneath a curve on a graph. This method is also known as taking an integral. There are 3 forms of Riemann Sums: Left, Right, and Middle.

B A To find the intervals needed, use the formula: Where B = the upper limit, A = the lower limit, and N = the number of rectangles used. N = 4

Then incorporate the previous intervals into the formula:

For a Left Riemann, use all of the functions except for the last one. The Left Riemann under approximates the area under the curve.

For a Right Riemann, use all of the functions except for the last one. The Right Riemann over approximates the area under the curve.

For a Middle Riemann, average all the intervals found and plug the averages into the functions. The Middle Riemann is the closest approximation.

The Middle Riemann is the closest approximation

N = 4

1.Click the “PRGM” button. 2.Select the RIEMANN program. 3.Enter your f(x). 4.Enter Lower & Upper bounds. 5.Enter Partitions 6.Select Left, Right, or Midpoint Sum

Like Riemann Sums, Trapezoidal Rule approximates the are under the curve using trapezoids instead of rectangles to better approximate.

Use the same formula to find your intervals. Then plug your intervals into the equation:

N = 4

Remember to multiply all intervals by 2, excluding the first and last interval.

N = 4

1.Click the “PRGM” button. 2.Select the RIEMANN program. 3.Enter your f(x). 4.Enter Lower & Upper bounds. 5.Enter Partitions 6.Select Trapezoid Sum

Simpson’s rule, created by Thomas Simpson, is the most accurate approximation of the area under a curve as it uses quadratic polynomials instead of rectangles or trapezoids.

Simpson’s Rule can ONLY be used when there are an even number of partitions. Still use the formula: to find your intervals to plug into the equation.

N = 4

When using Simpson’s Rule, multiply all intervals excluding the first and the last alternately between 4 & 2, always starting with 4

1.Click the “PRGM” button. 2.Select the SIMPSON program. 3.Enter Lower & Upper bounds. 4.Enter your N/2 Partitions. 5.Enter your f(x)