### Riemann Sums

Graphs a function f(x) and approximations to the area under the graph of f(x) for a given interval, using sums of areas of rectangles with the heights of the rectangles determined by f(x).
How to use   ||   Examples   ||   Other Notes

How to use
• Enter the function f (x) in the text input field marked "f (x)=" (Example: f (x)=4-x2)
• Click the "Graph" button (this button also refreshes the graph)
• Enter the endpoints of the interval [a,b] for the definite integral in the text input fields marked "a=" and "b=", and click the "Graph" button to refresh (Example: [a,b]=[-2,2])
• Use the "+" and "-" buttons under the label marked "n=" to change the number of subintervals used for the approximation (Example: n=8)
• The slider determines which point in each subinterval is used to determine the height of the rectangle, from left endpoint to right endpoint (Example: left endpoint)
• The "Left", "Mid", and "Right" buttons specify left endpoint, midpoint, and right endpoint, respectively, and adjust the slider appropriately
• To erase the graph and all input fields, setting the slider to a default position and n to a default value, click the "Clear" button
The text input field for f (x) can accept a wide variety of expressions to represent functions, and the buttons under the graph allow various manipulations of the graph coordinates. The text input fields for a and b can accept real numbers in decimal notation.
Examples
 Polynomial: f (x)=4-x2 [a,b]=[-2,2] n=20 for both    left|mid|right Exponential: f (x)=e2 [a,b]=[0,2] n=10 for both    left|mid|right Trigonometric: f (x) = sin x [a,b]=[0,3.14159] n=6 for both left|mid|right

Other Notes
The graph of f (x) is shown, with the rectangles over subintervals of [a,b] drawn in outline so that positive areas are green and negative areas are red. The approximate area computed is shown in the label marked "Area=".