### Substitution 1

Graphs a function f (x), the area under the graph of f (x) for a given interval, and the modifications made to f (x) and the area by a substitution u=g(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)=2x/(x2+1)).
• Click the "Graph" button (this button refreshes both graphs) -- at this point, the left graph will show the graph of f (x), and the right graph will remain empty.
• Enter the substitution function g(x) in the text input field marked "u=g(x)=", and its derivative g'(x) in the text input field marked "u'=g'(x)=". Then click the "Graph" button to refresh. Now the right graph will show the function transformed by the substitution. (Example: u=x2+1, u'=2x).
• 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]=[1,2]).
• To erase the graph and all input fields, setting a and b to default values, click the "Clear" button
The text input fields for 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 computing the derivative g'(x), try the Derivative Calculator.

Examples

Other Notes
The area under the graph of f (x) is drawn with the area under the curve on the interval [a,b] shaded so that positive areas are blue or green and negative areas are red or orange. The graph on the left shows the areas after making the substitution u=g(x), so plotting (g(x), f (x)/g'(x)) for x in the interval [a,b]. Correspondingly colored regions in the two graphs have equal area. Note that green and orange regions in the right graph have opposite orientation to their counterparts in the left graph. The endpoints of the interval in the right graph are given as c=g(a) and d=g(b) -- the values of c and d are shown in labels under the graph.

Be aware that regions in the left graph may overdraw (for example, a blue region may overdraw an orange region) -- regions are overdrawn as though they are semitransparent.