The Definite Integral

A definite integral quite simply returns the area under the graph of a function - the area between the graph of a function and the x - axis - between two x values.

The diagram shows the graph of
$y=4-x^2$
.
The curve intersects the x - axis at
$x=-2$
and
$x=2$
.
The area between the curve and the x -axis between the x values -2 and 2 is
$\int^2_{-2} x^2 dx = [\frac{x^3}{3}]^2_{-2} = \frac{2^3}{3}- \frac{(-2)^3}{3}=\frac{8}{3}- (- \frac{8}{3})=\frac{16}{3}$