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{{Math-h}}
A '''definite integral''' is an [[integral]] with upper and lower limits.
A '''definite integral''' is an [[integral]] with upper and lower limits.
==Definite Integrals==
==Definite Integrals==
A definite integral is the area under the curve between two points on the function. In the picture below, the yellow area is "positive" and the blue area is "negative". The integral is evaluated by adding the positive area together and subtracting the negative area.
A definite integral is the area under the curve between two points on the function. In the picture below, the yellow area is "positive" and the blue area is "negative". The integral is evaluated by adding the positive area together and subtracting the negative area.
If the function f(x) is real rather than complex, then the definite integral is also known as a Riemann integral.
If the function f(x) is real rather than complex, then the definite integral is also known as a Riemann integral.
== Solving Definite Integrals ==
== Solving Definite Integrals ==
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==
== 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>F(5) = {1 \over 3}5^3 = {125 \over 3}</math>
: <math>F(5) = {1 \over 3}5^3 = {125 \over 3}</math>
: <math>F(-3) = {1 \over 3}(-3)^3 = {-27 \over 3}</math>
: <math>F(-3) = {1 \over 3}(-3)^3 = {-27 \over 3}</math>
<br />And subtract:
: <math>{125 \over 3} - {-27 \over 3} = {125 \over 3} + {27 \over 3} </math>
: <math>{125 \over 3} - {-27 \over 3} = {125 \over 3} + {27 \over 3} </math>
: <math>= {152 \over 3}</math>
: <math>= {152 \over 3}</math>
: <math>\int_4^{12}\frac{3x+11}{x^2-x-6}dx </math>
: <math>\int_4^{12}\frac{3x+11}{x^2-x-6}dx </math>
=== Integration ===
=== Integration ===
<br />See the [[Partial fractions in integration]] page for how to integrate.
<br />See the [[Partial fractions in integration]] page for how to integrate the above expression (it is the example).
<br />As shown on the page mentioned above:
<br />As shown on the page mentioned above:
: <math>\int\frac{3x+11}{x^2-x-6}dx=4ln|x-3|-ln|x+2|+c</math>
: <math>\int\frac{3x+11}{x^2-x-6}dx=4ln|x-3|-ln|x+2|+c</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]]