Integrals: the basics

Signed area

A definite integral is a signed area: it is negative on intervals where the graph lies below the x-axis.

f(x)≤0 on [a,b]⇒∫abf(x) dx≤0f(x)\le 0 \text{ on } [a,b] \Rightarrow \int_a^b f(x)\,dx\le 0

A definite integral is a signed area. On intervals where the graph is below the xx-axis, the value is negative.

To find the actual area, split the interval at the points where the graph crosses the xx-axis, flip the sign of the negative parts, and add.

xyy = x² − 102
The part below the x-axis (blue) counts as negative. For the actual area, split at the crossing and flip the sign.Drag the dot

Examples

  1. 01What is the value of this definite integral?∫−22x3 dx\displaystyle \int_{-2}^{2} x^{3}\,dxAnswerHide
    Answer0\displaystyle 0
  2. 02What is the value of this definite integral?∫−33x3 dx\displaystyle \int_{-3}^{3} x^{3}\,dxAnswerHide
    Answer0\displaystyle 0
  3. 03What is the value of this definite integral?∫−11x3 dx\displaystyle \int_{-1}^{1} x^{3}\,dxAnswerHide
    Answer0\displaystyle 0
  4. 04What is the value of this definite integral?∫−11x1 dx\displaystyle \int_{-1}^{1} x^{1}\,dxAnswerHide
    Answer0\displaystyle 0
  5. 05What is the value of this definite integral?∫−33x1 dx\displaystyle \int_{-3}^{3} x^{1}\,dxAnswerHide
    Answer0\displaystyle 0
  6. 06What is the value of this definite integral?∫−22x1 dx\displaystyle \int_{-2}^{2} x^{1}\,dxAnswerHide
    Answer0\displaystyle 0

FAQ

Q1When is a definite integral negative?Show answerHide
A
When the graph is below the x-axis on that interval (f(x)≤0f(x)\le 0). If you are asked for the area, fix the sign of that part and add it.

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Before this
Definite integrals: F(b) − F(a)Sign of f' and increasing/decreasing
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Signed area
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Area between two curves

Published 2026-09-03 · Updated 2026-09-05