Difference between revisions of "Definite integral"
Jump to navigation
Jump to search
(Added examples and expanded article) |
DavidB4-bot (talk | contribs) (clean up & uniformity) |
||
| (8 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| + | {{Math-h}} | ||
| + | |||
A '''definite integral''' is an [[integral]] with upper and lower limits. | A '''definite integral''' is an [[integral]] with upper and lower limits. | ||
| Line 4: | Line 6: | ||
==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. | ||
| Line 10: | Line 14: | ||
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> | ||
| Line 29: | Line 60: | ||
: <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> | ||
| Line 36: | Line 68: | ||
: <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> | ||
| Line 47: | Line 79: | ||
: <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 | + | : <math>\approx 7.94160...</math> |
<br />Note the following: | <br />Note the following: | ||
* <math>ln(1) = 0</math> | * <math>ln(1) = 0</math> | ||
| Line 53: | Line 85: | ||
* <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 | + | == See also == |
*[[Integral]] | *[[Integral]] | ||
*[[Indefinite integral]] | *[[Indefinite integral]] | ||
| − | |||
| − | [[Category: | + | [[Category:Calculus]] |
| + | |||
| + | [[Category:Integration]] | ||
Latest revision as of 11:32, July 13, 2016
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} \sin x=?\,} | This article/section deals with mathematical concepts appropriate for late high school or early college. |
A definite integral is an integral with upper and lower limits.
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.
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 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 )
- The rule for integration is simply unknown. (Such as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \zeta(x)\,dx} , the Zeta function)
Example 1
|
| A graph representation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} 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:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-3}^{5} x^2 dx}
Integration
Using indefinite integration, it can be shown that:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x^2dx = {1 \over 3} x^3 = F(x)}
Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)}
is the indefinite integral of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)}
.
Subtraction
Now, plug Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3}
into the new expression and subtract, as shown by the Fundamental Theorem of Calculus.
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(5) = {1 \over 3}5^3 = {125 \over 3}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(-3) = {1 \over 3}(-3)^3 = {-27 \over 3}}
And subtract:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {125 \over 3} - {-27 \over 3} = {125 \over 3} + {27 \over 3} }
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = {152 \over 3}}
Example 2
This is a more complex definite integral that requires partial fractions to solve:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_4^{12}\frac{3x+11}{x^2-x-6}dx }
Integration
See the Partial fractions in integration page for how to integrate the above expression (it is the example).
As shown on the page mentioned above:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{3x+11}{x^2-x-6}dx=4ln|x-3|-ln|x+2|+c}
Subtraction
This means that we can now subtract:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left [ 4ln|12-3|-ln|12+2|+c \right ] - \left [ 4ln|4-3|-ln|4+2|+c \right ]}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left [ 4ln|9|-ln|14|+c \right ] - \left [ 4ln|1|-ln|6|+c \right ]}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 4ln|9|-ln|14| - 4ln|1| + ln|6|}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 4ln|9|-ln|14| + ln|6|}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \approx 7.94160...}
Note the following:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ln(1) = 0}
- The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} on each side cancels out because we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c-c}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ln|6|} becomes positive because it was Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -(-ln|6|)}