In calculus notation, this is written

The above is a definite integral which gives the area between the curve and the -axis. Integration is, in fact, the opposite of differentiation, and an indefinite integral gives the antiderivative of the gradient function, i.e. , where is a constant. The constant of integration is necessary since a constant always differentiates to . Also . As differentiation, integration is linear, i.e.

To evaluate a definite integral, we use the fundamental theorem of calculus which states that if ,

The indefinite integral evaluated at gives the area between and , where is a point depending on the integrand. Hence,

This justifies the previous statement. (This is obvious geometrically.)

To derive standard integrals, we simply reverse the differentiation
process. For example

Integrating gives

(since integration is also linear) so that

Other results can be derived in a similar way. To find the area between the curve , the -axis, and the lines and , we need to evaluate

This is done as follows