### Areas Between Two Curves

Graphs two functions, f(x) and g(x), and the area between the graphs of these functions for a given interval, and computes the area using given antiderivatives.
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)=x3-x)
• 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)=x4-x2/2)
• Enter the function g(x) in the text input field marked "g(x)=", and click the "Graph" button to refresh. (Example: g(x)=x)
• Enter the antiderivative G(x) (so that G '(x)=g(x)) in the text input field marked "G(x)=" (Example: G(x)=x2/2)
• 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])
• To erase the graph and all input fields, click the "Clear" button
The text input fields for f (x), F (x), g(x), and G(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 antiderivatives F (x) and G(x), try computing the derivatives F '(x) and G '(x) using the Derivative Calculator, and checking F '(x)=f (x) and G '(x)=g(x).

Examples
 Polynomial: f (x)=x3-x F (x)=x4/4-x2/2     g(x)=x G(x)=x2/2 [a,b]=[-2,2] Trigonometric: f (x)= sin x F (x)= cos x g(x)= cos x G(x)= -sin x [a,b]=[0.7854,2.3562]

Other Notes
The graph shows f (x) and g(x), with the area between the curves 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 between the curves.

In the "Polynomial" example above, the regions shaded have part with f (x) above g(x) (in blue) and part reversed (in red). The two areas are exactly the same, giving a value of zero for net area (blue area minus red area).