Difference between revisions of "Integral Command"

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(* in some versions of GeoGebra, a numerical algorithm is used so integrating up to an asypmtote or similar eg <code>Integral(ln(x), 0, 1)</code> won't work. In this case try <code>Integral(ln(x), 0, 1, false)</code>)
 
(23 intermediate revisions by 7 users not shown)
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<noinclude>{{Manual Page|version=4.0}}[[Category:Manual (official)|{{PAGENAME}}]]</noinclude>
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<noinclude>{{Manual Page|version=5.0}}</noinclude>{{command|cas=true|function}}
{{command|cas=true|function}}
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;Integral( <Function> )
;Integral[Function]
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:Gives the indefinite integral with respect to the main variable.
: Yields the indefinite integral for the given function with respect to the main variable.
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:{{example|1=<code><nowiki>Integral(x^3)</nowiki></code> yields <math>x^4 \cdot 0.25</math>.}}
:{{example|1=<div><code><nowiki>Integral[x³]</nowiki></code> yields ''  x⁴ / 4 ''.</div>}}
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;Integral( <Function>, <Variable> )
;Integral[ <Function>, <Variable> ]
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:Gives the partial integral with respect to the given variable.
:Returns the partial integral of the function with respect to the given variable.
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:{{Example|1=<code><nowiki>Integral(x³+3x y, x)</nowiki></code> gives'' <math>\frac{1}{4}x^4</math> + <math>\frac{3}{2}</math> x² y '' .}}
:{{example|1=<div><code><nowiki>Integral[x³+3x y, x]</nowiki></code> yields '' (x² (x² + 6y)) / 4 ''.</div>}}
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;Integral( <Function>, <Start x-Value>, <End x-Value> )
; Integral[Function, Number a, Number b]
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:Gives the definite integral over the interval ''[Start x-Value , End x-Value]'' with respect to the main variable.
: Returns the definite integral of the function, with respect to the main variable, in the interval [''a , b''].
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:{{note| 1=This command also shades the area between the function graph of ''f'' and the ''x''-axis.}}
: {{Note| This command also shadows the area between the function graph of ''f'' and the ''x''-axis.}}
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;Integral( <Function>, <Start x-Value>, <End x-Value>, <Boolean Evaluate> )
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:Gives the definite integral of the function over the interval ''[Start x-Value , End x-Value]'' with respect to the main variable and shades the related area if ''Evaluate'' is ''true''.  In case ''Evaluate'' is ''false'' the related area is shaded but the integral value is not calculated.
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==CAS Syntax==
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In the [[File:Menu view cas.svg|link=|16px]] [[CAS View]] undefined variables are allowed as input as well.
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:{{example|1=<code><nowiki>Integral(cos(a t), t)</nowiki></code> yields <math>\frac{sin(a t)}{a} + c_1</math>.}}
  
; Integral[Function, Number a, Number b, Boolean Evaluate]:
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Furthermore, the following command is only available in the [[File:Menu view cas.svg|link=|16px]] ''CAS View'':
: Returns the definite integral of the function, with respect to the main variable, in the interval [''a , b''] and shadows the related area when ''Evaluate = true''.  In case ''Evaluate = false'' the related area is shaded but the integral value is not calculated.
 
  
==CAS Syntax==
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;Integral( <Function>, <Variable>, <Start x-Value>, <End x-Value> )
; Integral[ Function f]
+
:Gives the definite integral over the interval ''[Start x-Value , End x-Value]'' with respect to the given variable.
: Yields the indefinite integral for the given function with respect to the main variable.
+
:{{example|1=<code><nowiki>Integral(cos(t), t, a, b)</nowiki></code> yields <math>- sin(a) + sin(b)</math>.}}
:{{Example|1=<code><nowiki>Integral[cos(x)]</nowiki></code> returns sin(x)+c1.}}
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; Integral[Function f, Variable t]
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{{note| 1=<div>
: Returns the indefinite integral of the function with respect to the given variable ''t''.
+
* The answer isn't guaranteed to be continuous, eg <code>Integral(floor(x))</code>, that is the integral of the function ⌊x⌋  - in that case you can define your own function to use eg <code>F(x)=(floor(x)² - floor(x))/2 + x floor(x) - floor(x)²</code>, i.e. the function <math>\frac{⌊x⌋² - ⌊x⌋}{2} + x \cdot⌊x⌋ - ⌊x⌋²</math>
:{{Example|1=<code><nowiki>Integral[cos(a t), t]</nowiki></code> returns sin(a t)/a+c2.}}
+
* in some versions of GeoGebra, a numerical algorithm is used so integrating up to an asypmtote or similar eg <code>Integral(ln(x), 0, 1)</code> won't work. In this case try <code>Integral(ln(x), 0, 1, false)</code>
; Integral[Function, Number a, Number b]
+
</div>}}
: Returns the definite integral of the function, with respect to the main variable, in the interval [''a , b''].
 
:{{Example|1=<code><nowiki>Integral[cos(x), a, b]</nowiki></code> returns sin(b) - sin(a).}}
 
; Integral[Function f, Variable t, Number a, Number b]
 
: Returns the definite integral in the interval [''a , b''] with respect to the given variable ''t''.
 
:{{Example|1=<code><nowiki>Integral[cos(t), t, a, b]</nowiki></code> returns sin(b) - sin(a).}}
 

Latest revision as of 12:11, 20 March 2019


Integral( <Function> )
Gives the indefinite integral with respect to the main variable.
Example: Integral(x^3) yields x^4 \cdot 0.25.
Integral( <Function>, <Variable> )
Gives the partial integral with respect to the given variable.
Example: Integral(x³+3x y, x) gives \frac{1}{4}x^4 + \frac{3}{2} x² y .
Integral( <Function>, <Start x-Value>, <End x-Value> )
Gives the definite integral over the interval [Start x-Value , End x-Value] with respect to the main variable.
Note: This command also shades the area between the function graph of f and the x-axis.
Integral( <Function>, <Start x-Value>, <End x-Value>, <Boolean Evaluate> )
Gives the definite integral of the function over the interval [Start x-Value , End x-Value] with respect to the main variable and shades the related area if Evaluate is true. In case Evaluate is false the related area is shaded but the integral value is not calculated.

CAS Syntax

In the Menu view cas.svg CAS View undefined variables are allowed as input as well.

Example: Integral(cos(a t), t) yields \frac{sin(a t)}{a} + c_1.


Furthermore, the following command is only available in the Menu view cas.svg CAS View:

Integral( <Function>, <Variable>, <Start x-Value>, <End x-Value> )
Gives the definite integral over the interval [Start x-Value , End x-Value] with respect to the given variable.
Example: Integral(cos(t), t, a, b) yields - sin(a) + sin(b).


Note:
  • The answer isn't guaranteed to be continuous, eg Integral(floor(x)), that is the integral of the function ⌊x⌋ - in that case you can define your own function to use eg F(x)=(floor(x)² - floor(x))/2 + x floor(x) - floor(x)², i.e. the function \frac{⌊x⌋² - ⌊x⌋}{2} + x \cdot⌊x⌋ - ⌊x⌋²
  • in some versions of GeoGebra, a numerical algorithm is used so integrating up to an asypmtote or similar eg Integral(ln(x), 0, 1) won't work. In this case try Integral(ln(x), 0, 1, false)
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