Integration and differentiation both have uses in finding information about graphs of functions. We already know that the derivative of a function gives the slope of the function. The definite integral between two points gives the area between the function and the x-axis.

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As an example, look at this graph of the function _{}. If I wanted to
find the area between the curve and the x-axis (x-axis located at y = 0)
between say x = –4 and x = –2, I would take the following definite integral:

_{}

Note that the left-most point (x = –4) is put on the bottom of the integral sign, and the right-most point (x = –2) is put on the top.

The indefinite integral is:

_{}

I substitute the top number (–2) on the integral sign
into this indefinite integral* *and get:

_{}

Then I substitute the bottom number (–4) on the integral sign into the indefinite integral and get:

_{}

I subtract the second result from the first:

_{}

This is the area between the curve and the x-axis between x = –4 and x = –2.