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{{Template:Math-h}}
{{Math-h}}
A '''definite integral''' is an [[integral]] with upper and lower limits.
A '''definite integral''' is an [[integral]] with upper and lower limits.
Solving a definite integral usually has two main steps: [[integration]] and [[subtraction]].
Solving a definite integral usually has two main steps: [[integration]] and [[subtraction]].
Sometimes approximations, such as the Riemann Integral or [[Simpson's rule]] are used. These approximations are used when:
*The exact answer is not needed, only a close approximation. (Common in [[Engineering]])
*The rule for integration is very complex. (Such as <math>\int e^{x^2}\,dx</math>)
*The rule for integration is simply unknown. (Such as <math>\int \zeta(x)\,dx</math>, the [[Zeta function]])
== Example 1==
{| class="infobox" style="text-align:center;"
<table class="infobox" style="float: right; text-align: center; font-size: 100%; clear:right; background:white; border: 2px solid blue;" cellspacing="0" cellpadding="2" width="33%">
|-
|[[Image:Firstexampledintegralsqa9.png|300px|Click on the image to get a better view.]]
|-
|A graph representation of <math>f(x)</math> in Example 1.
|-
|The ''thin blue line'' is f(x).
|-
|The ''yellow'' is what we add to get the value of the integral. It is the '''area'''.
|-
|The ''vertical green line'', the '''lower bound''', is at -3 because the lower bound of the definite integral is located at -3.
|-
|Likewise, the ''red line'' is the '''upper bound''', which is at 5.
|-
|Click on the image to get a better view.
|}
This is a very simple definite integral:
This is a very simple definite integral:
:<math>\int_{-3}^{5} x^2 dx</math>
:<math>\int_{-3}^{5} x^2 dx</math>
: <math>= 4ln|9|-ln|14| - 4ln|1| + ln|6|</math>
: <math>= 4ln|9|-ln|14| - 4ln|1| + ln|6|</math>
: <math>= 4ln|9|-ln|14| + ln|6|</math>
: <math>= 4ln|9|-ln|14| + ln|6|</math>
: <math>\approx 6.02941...</math>
: <math>\approx 7.94160...</math>
<br />Note the following:
<br />Note the following:
* <math>ln(1) = 0</math>
* <math>ln(1) = 0</math>
* <math>ln|6|</math> becomes positive because it was <math>-(-ln|6|)</math>
* <math>ln|6|</math> becomes positive because it was <math>-(-ln|6|)</math>
== See Also ==
== See also ==
*[[Integral]]
*[[Integral]]
*[[Indefinite integral]]
*[[Indefinite integral]]
[[Category:integration]]
[[Category:Calculus]]
[[Category:Integration]]