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)=x^{3}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)=x^{4}x^{2}/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)=x^{2}/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)=x^{3}x
F (x)=x^{4}/4x^{2}/2
g(x)=x
G(x)=x^{2}/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).