Mathlets: Definite Integrals

Definite Integrals

Graphs the area under a function over an interval, and computes the area using the antiderivative of the function.
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)=x²-1)
Click the "Graph" button (this button also refreshes the graph)
Enter the antiderivative F (x) (so that F '(x)=f (x)) in the text input field marked "F (x)=" (Example: F (x)=x³/3-x)
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]=[0,2])
To erase the graph and all input fields, click the "Clear" button

The text input fields for f (x) and 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.

For assistance checking the antiderivative F (x), try computing its derivative F '(x) and checking F '(x)=f (x) using the Derivative Calculator.

Examples

Polynomial:
f (x)=x²-1
F (x)=x³/3-x
[a,b]=[0,2]

Exponential:
f (x)=e^x
F (x)=e^x
[a,b]=[0,1]

Trigonometric:
f (x) = cos x
F (x) = sin x
[a,b]=[0,3]

Other Notes
The graph shows f(x), with the area under the curve on the interval [a,b] shaded so that positive areas are blue and negative areas are red. A label under the graph shows the net area under the graph.

In the "Polynomial" example above, part of the shaded area is above the x-axis (in blue) and part below it (in red). The area above the axis is a bit larger than the area below it, giving a positive value for net area (blue area minus red area).