Science Library - free educational site

Definite integrals

Fundamental Theorem of Calculus

If $f$ is a continuous function on the interval $a ≤ x ≤ b$ and $F$ is an antiderivative of $f$ on $a ≤ x ≤ b$, then:

$$∫↙{a}↖{b} f(x) dx = [F(x)]_a^b = F(b) - F(a)$$

Definite Integral Properties

  1. $∫↙{a}↖{b} kf(x)dx = k∫↙{a}↖{b} f(x)dx$
  2. $∫↙{a}↖{b} (f(x)±g(x))dx = ∫↙{a}↖{b} f(x)dx ± ∫↙{a}↖{b} g(x)dx$
  3. $∫↙{a}↖{a} f(x)dx = 0$
  4. $∫↙{a}↖{b} f(x)dx = -∫↙{b}↖{a} f(x)dx$
  5. $∫↙{a}↖{b} f(x)dx = ∫↙{a}↖{c} f(x)dx + ∫↙{c}↖{b} f(x)dx$
  6. $∫u⋅dv = uv - ∫v⋅du$

Definite Integral Properties

  • Constant Rule: $∫ kdx = kx + C$
  • Power Rule: $∫x^n dx = 1/(n + 1) x^{n+1} + C$, $n ≠ 1$
  • Constant multiple rule: $∫kf(x)dx = k ∫f(x) dx$
  • Sum of difference rule: $∫(f(x) ± g(x))dx = ∫f(x)dx ± ∫g(x)dx$
  • $e^x$ and $1/x$ integrals: $1/{x}dx = lnx + C$, $x > 0$, $∫e^xdx = e^x + C$
  • Linear composition: $∫f(ax + b)dx = 1/{a}F(ax + b) + C$, where $F'(x) = f(x)$


Two common methods for solving integrals are:

Substitution: ${dy}/{dx} = {dy}/{du}⋅{du}/{dx}$

Rearranging the equation to obtain the form $∫f(g(x))g'(x)dx$.

An indefinite integral is a family of functions that differ by the constant C. A definite integral is of the form:

$$∫↙a↖b f(x) dx$$

Leibniz invented the ∫ symbol for integration by the 1680s.

$∫↙a↖b f(x) dx$ is read as 'the integral from a to b of f(x) with respect to x'.

If a function $f$ is defined for a ≤ x ≤ b and ${lim}↙{n→∞}∑↙{i=1}↖n f(x_i)Δx_i$ exists, then $f$ is integrable on a ≤ x ≤ b.

This is the definite integral and it is denoted as $∫↙a↖b f(x) dx$, where a and b are the lower and upper limits of integration.

If $f$ is a non-negative function for a ≤ x ≤ b, then $∫↙a↖b f(x) dx$ gives the area under the curve from x = a to x = b.

Have you seen the C?

Definite integrals resolve to a number. This is because the constant C which is in the indefinite integral solution can be cancelled during the procedure of quantifying the definite integral.

The integral of a function
The definite integral from a to b of a function is the area between the curve from a to b and the x-axis.

From the graph of the function $f(x) = 4 - x^2$, determine the integral and use this to calculate the area under the curve from x = -2 to x = +2.

$A = ∫↙{-2}↖{2} f(x) dx = ∫↙{-2}↖{2} (4 - x^2) dx $
$= [4x - {x^3}/3 + C]_{-2}^{2} $
$= (8 - 8/3 + C) - (-8 + 8/3 + C) $
$= 16 - {16}/3 + C - C $
$= {32}/3 $
$= 10.67$ 

Area between two curves

If $y_1$ and $y_2$ are continuous on $a ≤ x ≤ b$ for all x in $a ≤ x ≤ b$, then the area between $y_1$ and $y_2$ from x = a to x = b is given by:
$$∫↙{a}↖{b} (y_1 - y_2)dx $$

Area under the x-axis

The area of a curve is never negative. Therefore, the part of a function which graphs to below the x-axis needs to be calculated separately from the part above the x-axis, and made positive.

If $f(x)$ is negative between $n$ and $p$, the area between the curve of $f(x)$, the x-axis, and the lines $x = n$ and $x = p$ is:


Area between curves

If functions $f$ and $g$ are continuous in the interval [a, b], and $f(x) ≥ g(x) for the region $a ≤ x /le; b$, the the area between the two graphs is:

$$A = ∫_a^bf(x)dx - ∫_a^bg(x)dx = ∫_a^b(f(x)- g(x)dx$$

Associated Mathematicians:

  • Isaac Newton
  • Gottfried Leibnitz
  • Jakob Bernoulli
  • Johann Bernoulli (I)

Content © All rights reserved. Created : February 5, 2015 Last updated :December 13, 2015

Latest Item on Science Library:

The most recent article is:


View this item in the topic:

Vectors and Trigonometry

and many more articles in the subject:

Subject of the Week


Information Technology, Computer Science, website design, database management, robotics, new technology, internet and much more. JavaScript, PHP, HTML, CSS, Python, ... Have fun while learning to make your own websites with

Computer Science

Great Scientists

Carl Linnaeus

1707 - 1778

Carl Linnaeus was a prolific writer, publishing books, lavishly illustrated, throughout his life. Through his travels, studies and collections, he developed a system of taxonomic nomenclature which is the basis of the modern system.

Carolus Linnaeus

Quote of the day...

How wonderful that we have met a paradox. Now we have some hope of making progress.

ZumGuy Internet Promotions

IT information forum by Sean Bone