Concept preview

What you will be able to do

1. 01State the evaluation form of the Fundamental Theorem of Calculus: if $F$ is any antiderivative of a continuous function $f$, then $\displaystyle\int_a^b f(x)\,dx = F(b)-F(a)$, and explain why the particular antiderivative you choose does not change the answer.
2. 02Evaluate definite integrals of polynomials, $\sin x$ and $\cos x$, $e^x$, and $\dfrac1x$ by finding an antiderivative $F$ and computing the clean number $F(b)-F(a)$.
3. 03Apply the net change theorem $\displaystyle\int_a^b r(t)\,dt = R(b)-R(a)$ to read a definite integral of a rate as a total accumulated change, stated with correct units and sign.

Here is the statement in full. Suppose $f$ is continuous on the closed interval $[a,\,b]$, and suppose $F$ is any antiderivative of $f$ — that is, any function with $F\,'(x)=f(x)$. Then the definite integral of $f$ from $a$ to $b$ is the change in $F$ across the interval: $\int_a^b f(x)\,dx = F(b)-F(a).$ The bracket notation $\left[F(x)\right]_a^b$ is shorthand for exactly this instruction — evaluate $F$ at the top limit $b$, evaluate it at the bottom limit $a$, and subtract the second from the first. No rectangles, no limits of sums; just one antiderivative and one subtraction.