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

$$∫↙{a}↖{b} f(x) dx = [F(x)]_a^b = F(b) - F(a)$$- $∫↙{a}↖{b} kf(x)dx = k∫↙{a}↖{b} f(x)dx$
- $∫↙{a}↖{b} (f(x)±g(x))dx = ∫↙{a}↖{b} f(x)dx ± ∫↙{a}↖{b} g(x)dx$
- $∫↙{a}↖{a} f(x)dx = 0$
- $∫↙{a}↖{b} f(x)dx = -∫↙{b}↖{a} f(x)dx$
- $∫↙{a}↖{b} f(x)dx = ∫↙{a}↖{c} f(x)dx + ∫↙{c}↖{b} f(x)dx$
- $∫u⋅dv = uv - ∫v⋅du$

- 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.

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.

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 $ $= 10.67$

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:

$$∫↙{a}↖{b} (y_1 - y_2)dx $$

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:

$$|∫_n^pf(x)dx|$$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 © Andrew Bone. All rights reserved. Created : February 5, 2015 Last updated :December 13, 2015

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Physics is the science of the very small and the very large. Learn about Isaac Newton, who gave us the laws of motion and optics, and Albert Einstein, who explained the relativity of all things, as well as catch up on all the latest news about Physics, on ScienceLibrary.info.

1654 - 1705

Jakob Bernoulli was the first of long series of Bernoulli family members, all mathematicians, whose combined contributions to mathematics and physics is unequalled in history.

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