Evaluating definite integrals

You evaluate a definite integral with the fundamental theorem: find an antiderivative, put in the upper limit, put in the lower limit, subtract.

Rules

Fundamental theorem: $\int_{a}^{b} f (x) d x = F (b) - F (a)$
The constant $C$ cancels when you subtract
Swapped limits: $\int_{b}^{a} f d x = - \int_{a}^{b} f d x$
Area below the x-axis counts as negative; for areas split at the zeros

Your turn

Three problems on this topic. Type like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x).

Task 1 of 3Definite integrals

Task

Evaluate

\int_{- 2}^{1} (9 x^{2} - 4 x + 5) d x

Value of the integral

Write like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x), sin(x). Use brackets: e^(2x), not e^2x. Numbers also as 3/4 or e^2 - 1.

Worked example

Evaluate $\int_{- 2}^{0} (9 x^{2} + 4 x - 1) d x$

Worked solution

1.Antiderivative: $F (x) = 3 x^{3} + 2 x^{2} - x$
2.Fundamental theorem: $F (0) - F (- 2) = 0 - (- 14) = 14$

Typical mistake: Computing $F (a) - F (b)$ instead of $F (b) - F (a)$ , or dropping brackets when inserting negative limits.

Common questions

Is the integral the same as the area?

Only if the function is not negative on the interval. Where the curve lies below the x-axis the integral is negative. For the area, split at the zeros and add the absolute values.

Antiderivatives Integration by substitution Integration by parts

Evaluating definite integrals

Rules

Your turn

Is the integral the same as the area?

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