Library / Foundations: Maths / FMA-27

Foundations: Maths

Definite integral of a power

Area under y = xⁿ between two limits.

Ref: FMA-27
Rev: v1
Field: Foundations: Maths
Result: ∫

\int_{a} x n dx = \frac{b^{n + 1} - a^{n + 1}}{n + 1}

For n ≠ −1 the antiderivative of xⁿ is xⁿ⁺¹/(n+1) + C. Evaluate as F(b) − F(a). For total area split where the function changes sign.

Integration adds up a changing quantity; for xⁿ, raise the power by one, divide by the new power, and subtract the value at the lower limit from the value at the upper.

Calculator

Solve for

∫Definite integralComputed

aLower limit a

bUpper limit b

2.56

nPower n

Antiderivative at b9
Mean value over [a, b]4.333

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Definite integral () current value 8.667

bUpper limit bKey driver

4.8871.7

nPower n

448.4

aLower limit a

6.339

The integral is the signed area; areas below the axis count as negative.
Work is the integral of force over distance, and distance is the integral of speed over time.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
b Upper limit b	2.5 to 6	4.875	71.67	66.79	+10%	+34.4 %
n Power n	1 to 4	4	48.4	44.4	+10%	+17.7 %
a Lower limit a	0 to 2	6.333	9	2.667	+10%	−1.27 %

Sweep of Upper limit b

Upper limit b	Definite integral
2.5	4.875
2.938	8.116
3.375	12.48
3.813	18.14
4.25	25.26
4.688	34
5.125	44.54
5.563	57.04
6	71.67

Sweep one variable

Vary

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Variables and how each one contributes

∫

Definite integral

Result ·

What the equation gives you. Everything else feeds it.

Lower limit a

· typical 0 to 2

Where the integration starts. Raising a removes area from the left.

Upper limit b

· typical 2.5 to 6

Where the integration ends. Adds area on the right, strongly for large b.

Power n

· typical 1 to 4

The power of x. Higher powers make the right-hand end dominate.

Worked example with the values above

F(x) = x³/3.
F(b) − F(a) = 9 − 0.3333.
Integral = 8.667.

Assumptions and limits

n ≠ −1.
The function does not change sign between the limits for an area.

Common mistakes

Forgetting to divide by the new power.
Subtracting in the wrong order.

Statistics and uncertainty

Numerical estimates of the same integral (trapezium rule) improve as strips get narrower; the exact integral has no such error.

Where it is used

Work done and total change from a rate.
Area under a speed–time graph.

Maths you need

Powers: xⁿ
Differentiation: integration reverses it

Related formulas

Practise this formula