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Mathematical Expression Editor

At this point we have three “different” integrals.

At this point we have three different “integrals.” Let’s see if we can sort out the
differences.

Indefinite integrals

An indefinite integral, also called an antiderivative computes classes of functions:
Here there are no limits of integration, and your answer will have a “” at the end.
Pay attention to the notation:

Where .

Indefinite integrals havedo not have limits of integration, and they compute signed areaan antiderivativea class of
antiderivatives.

Two students, say Devyn and Riley, are working with the following indefinite
integral: Devyn computes the integral as and Riley computes the integral as Which
student is correct?

Devyn is correctRiley is correctBoth students are correctNeither student is correct

Both students are correct! The seeming discrepancy arises from the fact that the
“+C” in each case is different!

Accumulation functions

An accumulation function, also called an area function computes accumulated
area: This is a function of whose derivative is , with the additional property that .
Pay attention to the notation:

Where .

Accumulation functions havedo not have limits of integration, and they compute signed areaan antiderivativea class of
antiderivatives.

True or false: There exists a function such that

truefalse

Let this is an accumulation function and , since no area is accumulated yet. However,
. Hence there can be no such function . On the other hand, there is a function with
namely, . This subtlety arises from the fact that an accumulation function
gives a specific antiderivative of , the one that when evaluated at is zero.

Definite integrals

A definite integral computes signed area: Here we always have limits of
integration, both of which are numbers. Moreover, definite integrals have definite
values, the signed area between and the -axis. Pay attention to the notation:

Where .

Definite integrals havedo not have limits of integration, and they compute signed areaan antiderivativea class of
antiderivatives.