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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.
1 Indefinite integrals
An indefinite integral, also called an antiderivative computes classes of functions:
\[ \int f(x) dx = \text {``a class of functions whose derivative is $f$''} \]
Here
there are no limits of integration, and your answer will have a “\(+C\)” at the end. Pay
attention to the notation:
Where \(F'(x) = f(x)\).
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 is correct Riley is correct Both students are
correct Neither student is correct
Both students are correct! The seeming
discrepancy arises from the fact that the “+C” in each case is different!
2 Accumulation functions
An accumulation function, also called an area function computes accumulated
area:
\[ \int _a^x f(t) dt = \text {``a function $F$ whose derivative is $f$''} \]
This is a function of \(x\) whose derivative is \(f\), with the additional property that \(F(a)=0\).
Pay attention to the notation:
Where \(F'(x) = f(x)\).
Accumulation functions havedo not have limits of integration, and they
compute signed areaan antiderivativea class of antiderivatives.
True or
false: There exists a function \(f\) such that
\[ \int _0^x f(t) dt = e^x \]
true false
Let
\[ F(x) = \int _0^x f(t) dt, \]
this is an
accumulation function and \(F(0) = 0\), since no area is accumulated yet. However, \(e^0 =1\). Hence
there can be no such function \(f\). On the other hand, there is a function \(g\) with
\[ \int _0^x g(t) dt = e^x-1 \]
namely, \(g(x) = e^x\). This subtlety arises from the fact that an accumulation function
\[ F(x) = \int _a^x f(t) dt \]
gives a specific antiderivative of \(f\), the one that when evaluated at \(x=a\) is zero.
3 Definite integrals
A definite integral computes signed area:
\[ \int _a^b f(x) dx = \text {``the signed area between the $x$-axis and $f$''} \]
Here we always have limits of
integration, both of which are numbers. Moreover, definite integrals have
definite values, the signed area between \(f\) and the \(x\)-axis. Pay attention to the
notation:
Where \(F'(x) = f(x)\).
Definite integrals havedo not have limits of integration, and
they compute signed areaan antiderivativea class of antiderivatives.