Difference between revisions of "User:Temperal/The Problem Solver's Resource10"
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*<math>\int \csc \, dx =\ln |\csc x + \cot x| + c</math> | *<math>\int \csc \, dx =\ln |\csc x + \cot x| + c</math> | ||
*<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> |
+ | ===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. | ||
+ | ====Definition and Notation==== | ||
+ | *The definite integral of a function between <math>a</math> and <math>b</math> is written as <math>\int^{b}_{a}f(x)\,dx</math>. | ||
+ | *<math>\int^{b}_{a}f(x)\,dx=F(b)-F(a)</math>, where <math>F(x)</math> is the antiderivative of <math>f(x)</math>. This is also notated <math>\int f(x)\,dx \eval^{b}_{a}</math>, read as "The integral of <math>f(x)</math> evaluated at <math>a</math> and <math>b</math>." Note that this means in definite integration, one need not add a constant, as the constants from the functions cancel out. | ||
+ | ====Rules of Definite Integrals==== | ||
+ | *<math>\int^{b}_{a}f(x)\,dx=\int^{b}_{c}f(x)\,dx+\int^{c}_{a}</math> for any <math>c</math>. | ||
+ | ====Fundamental Theorem of Calculus==== | ||
+ | Let <math>{a}</math>, <math>{b} \in \mathbb{R}</math> , <math>a<b</math>. Suppose <math>F:[a,b] \to \mathbb{R}</math> is differentiable on the whole interval <math>[a,b]</math> (using limits from the right and left for the derivatives at <math>{a}</math> and <math>{b}</math>, respectively), and suppose that <math>F'</math> is Riemann integrable on <math>[a,b]</math>. Then <math>\int_a^b F'(x)dx = F(b) - F(a)</math>. | ||
+ | |||
+ | In other words, "the total change (on the right) is the sum of all the little changes (on the left)." | ||
[[User:Temperal/The Problem Solver's Resource9|Back to page 9]] | [[User:Temperal/The Problem Solver's Resource11|Continue to page 11]] | [[User:Temperal/The Problem Solver's Resource9|Back to page 9]] | [[User:Temperal/The Problem Solver's Resource11|Continue to page 11]] | ||
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Revision as of 17:35, 17 October 2007
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 |