Once you can find an integral, evaluating a definite integral stops being an area calculation and becomes something much shorter: substitute the two limits and subtract.

Definition The theorem

\int_{a}^{b} f(x)\,dx = F(b) - F(a)

Here $F$ is any integral of $f$ — a function whose derivative is $f$ . Evaluate it at the upper limit $b$ , evaluate it at the lower limit $a$ , and subtract the second from the first.

Find the integral, put in the top, put in the bottom, subtract.

Example The area under

f(x) = x

from 3 to 6

The integral of $x$ is $F(x) = \dfrac{x^{2}}{2}$ . Substitute $6$ , then subtract the value at $3$ .

F(6) = \dfrac{36}{2} = 18

F(3) = \dfrac{9}{2} = 4.5

Subtract:

18 - 4.5

⟹ $\displaystyle\int_{3}^{6} x\,dx = 13.5$

The region is a trapezium with parallel sides $3$ and $6$ and width $3$ , so its area is $\tfrac{1}{2}(3 + 6)(3) = 13.5$ . The theorem and the geometry agree.

Example An exponential integral

The integral of $e^{x}$ is $e^{x}$ itself, since $\ln e = 1$ . Nothing changes about the method.

F(x) = e^{x}

\displaystyle\int_{2}^{4} e^{x}\,dx = F(4) - F(2)

⟹ $e^{4} - e^{2}$

Leaving the answer as $e^{4} - e^{2}$ is exact; a decimal is only needed when a number is asked for.

Note Why the constant

C

drops out

An integral is usually written with $+ C$ . In a definite integral that constant never survives, because it sits in both terms and the subtraction removes it.

\bigl(F(b) + C\bigr) - \bigl(F(a) + C\bigr)

= F(b) - F(a) + C - C

= F(b) - F(a)

So the $+ C$ can simply be left off from the start when the integral is definite.

Note Reversing the limits

Always compute upper limit minus lower limit. Swapping the two limits swaps the sign of the result.

\int_{b}^{a} f(x)\,dx = -\int_{a}^{b} f(x)\,dx

Take $f(x) = 2x$ , with $F(x) = x^{2}$ :

\displaystyle\int_{1}^{3} 2x\,dx = F(3) - F(1) = 9 - 1 = 8

\displaystyle\int_{3}^{1} 2x\,dx = F(1) - F(3) = 1 - 9 = -8

Moving left to right across the region gives the area a positive sign; moving right to left gives the same size with a negative sign. The sign records the direction, not a different amount.

Summary

$\displaystyle\int_{a}^{b} f(x)\,dx = F(b) - F(a)$ , where $F$ is an integral of $f$ .
The method: find $F$ , substitute the upper limit, subtract the value at the lower limit.
$\displaystyle\int_{3}^{6} x\,dx = 18 - 4.5 = 13.5$ , matching the trapezium's area.
$\displaystyle\int_{2}^{4} e^{x}\,dx = e^{4} - e^{2}$ — same method, since the integral of $e^{x}$ is $e^{x}$ .
The $+ C$ cancels in the subtraction, so it is left off in a definite integral.
Swapping the limits negates the result: $\displaystyle\int_{b}^{a} = -\int_{a}^{b}$ .

The Fundamental Theorem of Calculus