Definite vs. Indefinite Integrals
This video goes over the difference between definite (bounded) and indefinite (unbounded) integrals.
Difference Between Definite ($\int$) and Indefinite Integral ($\int_a^b$)
An indefinite integral looks like
\[\int f(x)\,dx.\]A definite integral looks like
\[\int_a^b f(x)\,dx.\]Notice, an indefinite integral has no bounds on the integral sign, while a definite integral has a lower bound ($a$) and an upper bound ($b$).
The result of an indefinite integral is 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 indefinite integral example in the video is
\[\int x^2\,dx.\]There are no bounds on the integral sign, so we add a $+C$ to the antiderivative:
\[\int x^2\,dx=\frac{x^3}{3}+C.\]When we take the derivative of a function, constants disappear, so when we reverse the direction with an indefinite integral, we represent the constant we would gain back with $+C$.
Notice that for any value of $C$
\[\frac{x^3}{3}+C\]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:
\[\int_0^1 x^2\,dx.\]We get the same antiderivative we found for the indefinite integral:
\[\int x^2\,dx=\frac{x^3}{3}.\]We then evaluate the antiderivative at the top and bottom bounds (based on the Fundamental Theorem of Calculus):
\[\int_0^1 x^2\,dx= \left[\frac{x^3}{3}\right]_0^1.\]That means
\[\left[\frac{x^3}{3}\right]_0^1 =\frac{1^3}{3}-\frac{0^3}{3} =\frac{1}{3}.\]The $\frac{1}{3}$ we get represents the signed area under the curve $x^2$ from $x=0$ to $x=1$.
Watch this section: definite integral at 1:15.
The Fundamental Theorem of Calculus
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
\[F(1)-F(0)=\frac{1}{3}-0=\frac{1}{3}.\]Watch this section: Fundamental Theorem setup at 1:35.
Review From The Video
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,
\[\int_a^b f(x)\,dx=F(b)-F(a).\]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.