Definite integrals

5 minAS Level1.8.3

By the end of this lesson

By the end you can evaluate a definite integral, like $\displaystyle\int_1^3 (3x^2 - 2)\,dx$ , and get a number.

What they ask

Evaluate $\displaystyle\int_1^3 (3x^2 - 2)\,dx$ , or find the exact value of ….

The numbers are the limits: 3 is the upper limit (top) and 1 the lower limit (bottom).
The answer is a number, not a function of $x$ .

Integrate as usual, but skip the $+\,c$ .
Write the result in square brackets with the limits: $\Big[\;\ldots\;\Big]_1^3$ .
Put the top limit in, then subtract the bottom limit put in. Keep each in its own bracket, so the minus sign hits all of the second one.
Simplify. If they say exact, keep fractions and roots.

Why no + c?

Try keeping it:

$\Big[x^2 + c\Big]_0^2 = (4 + c) - (0 + c) = 4$

The $c$ cancels, and it always will.

So leave it out of definite integrals. It's still needed in every indefinite one (no limits).

Worked example: The easiest one: check it with a triangle

Evaluate $\displaystyle\int_0^2 2x\,dx$ .

Have a go first?

Worked example: Watch the bottom limit

Evaluate $\displaystyle\int_1^3 (3x^2 - 2)\,dx$ .

Have a go first?

Quick check

CalculateQuestion 1 · Recall

Multiple choiceQuestion 2 · Apply

Answer the 2 questions above to continue.