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# Integral calculus

Integral calculus is the study of areas under curves. If the area under a curve is split into strips, each of length , so the area of one of the strips is . Then the total area under the curve from to is

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

Subsections

Next: Methods of integration Up: Calculus - a brief Previous: Reciprocal rule
Alexander Frolkin 2001-03-13