Definite vs. Indefinite Integrals
This page follows the JoeCMath video Definite vs. Indefinite Integrals: What’s the Difference?. The video compares the two kinds of integral notation by using the same function, first without bounds and then with bounds from $0$ to $1$.
Main Difference
An indefinite integral has no bounds on the integral sign, while a definite integral has a lower bound and an upper bound.
An indefinite integral looks like
\[\int f(x)\,dx.\]A definite integral looks like
\[\int_a^b f(x)\,dx.\]An indefinite integral gives a family of functions. A definite integral gives one number that represents the signed area over the interval from $a$ to $b$.
Watch this section: introduction at 0:00.
Indefinite Integral
The first example in the video is
\[\int x^2\,dx.\]There are no bounds on the integral sign, so the result is an antiderivative:
\[\int x^2\,dx=\frac{x^3}{3}+C.\]The $+C$ matters because the answer represents a collection of functions. Any function of the form
\[\frac{x^3}{3}+C\]has derivative $x^2$, no matter which constant is chosen.
Watch this section: indefinite integral at 0:21.
Definite Integral
The definite integral example uses the same integrand, but now the integral has bounds:
\[\int_0^1 x^2\,dx.\]The antiderivative step is still needed:
\[\int x^2\,dx=\frac{x^3}{3}.\]Then the video evaluates that antiderivative at the top and bottom bounds:
\[\int_0^1 x^2\,dx= \left[\frac{x^3}{3}\right]_0^1.\]That means
\[\left[\frac{x^3}{3}\right]_0^1 =\frac{1^3}{3}-\frac{0^3}{3} =\frac{1}{3}.\]The final answer is not a function with $+C$. It is the number $\frac{1}{3}$.
Watch this section: definite integral at 1:15.
The Fundamental Theorem Step
The video shows the Fundamental Theorem of Calculus as the bridge between the antiderivative and the definite answer.
If $F$ is an antiderivative of $f$, then
\[\int_a^b f(x)\,dx=F(b)-F(a).\]For the video’s example, $F(x)=\frac{x^3}{3}$, $a=0$, and $b=1$, so
\[F(1)-F(0)=\frac{1}{3}-0=\frac{1}{3}.\]That is why a definite integral can use antiderivatives but still end as a number.
Watch this section: Fundamental Theorem setup at 1:35.
Checking The Difference Visually
The video then checks examples in WolframAlpha. The point of that section is not to change the method, but to make the output types easier to see.
For an indefinite integral, the output is an antiderivative family. For a definite integral, the graph can show a shaded region over the chosen interval, and the output is a numerical signed area.
The video also shows that signed area can be positive or negative depending on where the graph sits relative to the $x$-axis. Area above the axis contributes positively, and area below the axis contributes negatively.
Watch this section: WolframAlpha examples at 2:53.
Review From The Video
The closing review puts the two ideas side by side.
For an indefinite integral,
\[\int f(x)\,dx=F(x)+C.\]The result is a collection of functions, and $C$ is an arbitrary constant.
For a definite integral,
\[\int_a^b f(x)\,dx=F(b)-F(a).\]The result is one number. Geometrically, it represents signed area over the interval from $a$ to $b$.
Watch this section: review at 4:15.
Timestamp Guide
Related Calculus Work
The antiderivative step in this video uses the integration power rule.
For more on why indefinite integrals include $+C$, review the constant of integration notes.
You can also continue with u-substitution or the full JoeCMath integration playlist.