Difference between revisions of "Integral"
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*<math>\int^{b}_{a}f(x)\,dx=F(b)-F(a)</math>, where <math>F(x)</math> is the antiderivative of <math>f(x)</math>. This is also notated <math>\int f(x)\,dx \eval^{b}_{a}</math>, read as "The integral of <math>f(x)</math> evaluated at <math>a</math> and <math>b</math>." Note that this means in definite integration, one need not add a constant, as the constants from the functions cancel out. | *<math>\int^{b}_{a}f(x)\,dx=F(b)-F(a)</math>, where <math>F(x)</math> is the antiderivative of <math>f(x)</math>. This is also notated <math>\int f(x)\,dx \eval^{b}_{a}</math>, read as "The integral of <math>f(x)</math> evaluated at <math>a</math> and <math>b</math>." Note that this means in definite integration, one need not add a constant, as the constants from the functions cancel out. | ||
===Rules of Definite Integrals=== | ===Rules of Definite Integrals=== | ||
− | *<math>\int^{b}_{a}f(x)\,dx=\int^{b}_{c}f(x)\,dx+\int^{c}_{a}</math> for any <math>c</math>. | + | *<math>\int^{b}_{a}f(x)\,dx=\int^{b}_{c}f(x)\,dx+\int^{c}_{a}f(x)\,dx</math> for any <math>c</math>. |
==Other uses== | ==Other uses== |
Revision as of 22:55, 6 February 2008
The integral is one of the two base concepts of calculus, along with the derivative.
There are two types of integrals:
Contents
Indefinite Integral
The indefinite integral, or antiderivative, is a partial inverse of the derivative. That is, if the derivative of a function is written as
, then the indefinite integral of
is
, where
is a real constant. This is because the derivative of a constant is
.
Notation
- The integral of a function
is written as
, where the
means that the function is being integrated in relation to
.
- Often, to save space, the integral of
is written as
, the integral of
as
, etc.
Rules of Indefinite Integrals
for a constant
and another constant
.
,
Definite Integral
The definite integral is also the area under a curve between two points and
. For example, the area under the curve
between
and
is
, as are below the x-axis is taken as negative area.
Definition and Notation
- The definite integral of a function between
and
is written as
.
, where
is the antiderivative of
. This is also notated $\int f(x)\,dx \eval^{b}_{a}$ (Error compiling LaTeX. Unknown error_msg), read as "The integral of
evaluated at
and
." Note that this means in definite integration, one need not add a constant, as the constants from the functions cancel out.
Rules of Definite Integrals
for any
.
Other uses
- The word integral is the adjectival form of the noun "integer." Thus,
is integral while
is not.