Difference between revisions of "User:Temperal/The Problem Solver's Resource10"
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The there are two types of integrals: | The there are two types of integrals: | ||
===Indefinite Integral=== | ===Indefinite Integral=== | ||
− | The indefinite integral, or antiderivative, is a partial inverse of the derivative. That is, if the derivative of a function <math>f(x)</math> is written as <math>f'(x)</math>, then the indefinite integral of <math>f'(x)</math> is <math>f(x)+c</math>, where <math>c</math> is a real constant. This is because the | + | The indefinite integral, or antiderivative, is a partial inverse of the derivative. That is, if the derivative of a function <math>f(x)</math> is written as <math>f'(x)</math>, then the indefinite integral of <math>f'(x)</math> is <math>f(x)+c</math>, where <math>c</math> is a real constant. This is because the derivative of a constant is <math>0</math>. |
====Notation==== | ====Notation==== | ||
*The integral of a function <math>f(x)</math> is written as <math>\int f(x)\,dx</math>, where the <math>dx</math> means that the function is being integrated in relation to <math>x</math>. | *The integral of a function <math>f(x)</math> is written as <math>\int f(x)\,dx</math>, where the <math>dx</math> means that the function is being integrated in relation to <math>x</math>. | ||
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*<math>\int \cot x\,dx = \ln |\sin x| + c</math> | *<math>\int \cot x\,dx = \ln |\sin x| + c</math> | ||
*<math>\int cx\, dx=c\int x\,dx</math> | *<math>\int cx\, dx=c\int x\,dx</math> | ||
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===Definite Integral=== | ===Definite Integral=== | ||
The definite integral is also the area under a curve between two points <math>a</math> and <math>b</math>. For example, the area under the curve <math>f(x)=\sin x</math> between <math>-\frac{\pi}{2}</math> and <math>\frac{\pi}{2}</math> is <math>0</math>, as are below the x-axis is taken as negative area. | The definite integral is also the area under a curve between two points <math>a</math> and <math>b</math>. For example, the area under the curve <math>f(x)=\sin x</math> between <math>-\frac{\pi}{2}</math> and <math>\frac{\pi}{2}</math> is <math>0</math>, as are below the x-axis is taken as negative area. |
Revision as of 11:37, 27 June 2008
IntegralsThis section will cover integrals and related topics, the Fundamental Theorem of Calculus, and some other advanced calculus topics. The there are two types of integrals: Indefinite IntegralThe indefinite integral, or antiderivative, is a partial inverse of the derivative. That is, if the derivative of a function Notation
Rules of Indefinite Integrals
Definite IntegralThe definite integral is also the area under a curve between two points Definition and Notation
Rules of Definite Integrals
Fundamental Theorem of CalculusLet In other words, "the total change (on the right) is the sum of all the little changes (on the left)." Back to page 9 | Continue to page 11 |