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==<span style="font-size:20px; color: blue;">Integrals</span>==
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==<span style="font-size:20px; color: blue;">More Number Theory</span>==
This 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 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 derivative of a constant is <math>0</math>.
 
====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>.
 
*Often, to save space, the integral of <math>f(x)</math> is written as <math>F(x)</math>, the integral of <math>h(x)</math> as <math>H(x)</math>, etc.
 
====Rules of Indefinite Integrals====
 
*<math>\int c\,dx=cx+C</math> for a constant <math>c</math> and another constant <math>C</math>.
 
*<math>\int f(x)+g(x)...+z(x)\,dx=\int f(x)\,dx+\int g(x)\,dx...+\int z(x)\,dx</math>
 
*<math>\int x^n\,dx=\frac{1}{n+1}x^{n+1}+c</math>, <math>n \ne -1</math>
 
*<math>\int x^{-1}\,dx=\ln |x|+c</math>
 
*<math>\int \sin x\,dx = -\cos x + c</math>
 
*<math>\int \cos x\,dx = \sin x + c</math>
 
*<math>\int\tan x\,dx =  \ln |\cos x| + c</math>
 
*<math>\int \sec x\,dx = \ln |\sec x + \tan 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 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)."
 
  
 
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