Areas Between Two Curves
Graphs two functions, f
) and g
and the area between the graphs of these functions
for a given interval, and computes the area using given antiderivatives.
How to use
How to use
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.
- Enter the function f (x) in the text
marked "f (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)="
- Enter the function g(x) in the text
marked "g(x)=", and click the
"Graph" button to refresh.
- Enter the antiderivative G(x)
(so that G '(x)=g(x)) in the text input
field marked "G(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.
- To erase the graph and all input fields, click the
For assistance checking the antiderivatives
F (x) and
computing the derivatives F '(x) and
G '(x) using the
Derivative Calculator, and checking
F '(x)=f (x) and
f (x)= sin x
F (x)= cos x
g(x)= cos x
G(x)= -sin x
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
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
(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