How to evaluate a definite integral

A definite integral has two limits attached to the integral sign and returns a single number rather than a function. Where an indefinite integral asks 'what function differentiates to give this?', a definite integral asks 'what is the exact value between $x=a$ and $x=b$?'

That value is the signed area between the curve and the $x$-axis over the interval. In Add Math you use it to find areas under curves, volumes of revolution, and the total change in a quantity from its rate.

The rule that makes it work is the Fundamental Theorem of Calculus: integrate to get $F(x)$, then evaluate the difference between the top and bottom limits.

Use this method whenever the integral sign carries a lower and an upper limit, or whenever a question asks for a numerical value from an integration,'evaluate', 'find the area', or 'calculate the exact value of'. If there are no limits, you have an indefinite integral instead and must keep the $+c$.

The signal is the pair of numbers on the integral: $\int_{a}^{b}$. The moment you see them, plan for three actions, integrate, substitute both limits, subtract.

Because the answer is a number, you can and should check that it is reasonable: an area should normally be positive, and a definite integral of a simple polynomial over a short interval should give a tidy value. Because Add Math is marked analytically, showing the integrated expression and the substitution earns method marks even if the final arithmetic slips.

1. 1

   Integrate the function

   Integrate $f(x)$ term by term using $\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c$. For a definite integral you do not need the $+c$; it cancels in the subtraction.

2. 2

   Write the limit brackets

   Put the result in square brackets with the lower and upper limits, as $\big[F(x)\big]_{a}^{b}$.

3. 3

   Substitute the upper limit

   Replace $x$ with the top value $b$ to get $F(b)$.

4. 4

   Substitute the lower limit

   Replace $x$ with the bottom value $a$ to get $F(a)$.

5. 5

   Subtract

   Compute $F(b)-F(a)$, keeping the brackets so a negative $F(a)$ becomes a plus.

6. 6

   Simplify

   Work out the arithmetic to a single exact number, and state units if the context calls for them.

• Adding $+c$ to a definite integral, it always cancels, so it should not appear in the final line.
• Subtracting the wrong way round; the rule is $F(b)-F(a)$, upper minus lower.
• Dropping the brackets in $F(b)-F(a)$, so a negative $F(a)$ is subtracted with the wrong sign, here $4-(-2)$ is $6$, not $2$.
• Differentiating instead of integrating; the power rises by one, it does not fall.
• Substituting the limits before integrating, which mixes up the two operations.

The most costly slip we see is a sign error at the subtraction line, a student writes $4-2$ instead of $4-(-2)$ and loses the mark on the last step of an otherwise perfect solution. In one-to-one lessons our teachers have you keep the brackets around $F(a)$ every single time, and say 'top minus bottom' aloud before subtracting, so the habit protects you under exam pressure.

We also check that you never carry a stray $+c$ into a definite integral. Every teacher on spmaddmath.com.my is experienced.

Lessons are online and taught in English, paced to how quickly the method is settling for you.