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Integration
[Antiderivatives] [Indefinite Integration] [Indefinite Integration/Area Under a Curve] [Approximating Area Under a Curve] [U Substitution] [Integration Links]
Antiderivatives
- Notion of the antiderivative
- Just like it is possible to find the derivative of f, it is possible to find the antiderivative of f’. It is called indefinite integration. For example, if
 , then the derivative of that is  making the integral of that being  , which equals  *since the constant is unknown, the variable “C” is used*
Indefinite Integration
- There are several rules that help with finding the antiderivative in indefinite integration. Some of these rules include the following:
- example: integrate
- example: evaluate
Definite Integration/Area Under a Curve
- Just like differentiation is used to find the slope at a point, integration has an application. Integration can be used to find the area under a curve, like shown below:
- Using the Fundamental Theorem of Calculus, it is possible to find the area under a curve. The Fundamental Theorem of Calculus states
- If a function f is continuous on the closed interval
![[a,b]](ab1.jpg) and F is the antiderivative of f on the intervall , then
- Guidelines for using the Fundamental Theorem of Calculus are as follows:
- 1. Provided you can find an antiderivative of f, you now have a way to evaluate a definite integral without having to use the limit of a sum.
- 2. When applying the Fundamental Theorem of Calculus, the following notation is conventient.
- For instant, to evaluate
 , you can write
- 3. It is not neccessary to include a constant of integration C in the antiderivative because
- example: find the area under the curve of
Approximating Area Under a Curve
- In some cases, it is not easy or impossible to find the antiderivative of f. In those cases, it is only possible to approximate the area under the curve using one of several methods. The easiest and most accurate of these methods are the Trapezoidal Rule and Simpon's Rule. Just like with all approximations, these are simply very close approximations, not the exact area.
- The Trapezoidal Rule
- Let f be continuous on . The Trapezoidal Rule for approximating
 is given by
- Moreover, as
 , the right-hand side approaches .
- Simpson's Rule
- Let f be continuous on . Simpson's Rule for approximating is
- Moreover, as , the right-hand side approaches .
- Guidelines for using the Trapezoidal Rule and Simpon's Rule are as follows:
- 1. Determine how many times the interval will be split up.
- 2. Divide the interval into equal parts determined by what number n is as decided in the previous step.
- 3. Choose which rule to use (Simpson's Rule can only be used when n is even).
- 4. Plug in the values into the rule.
- 5. Compute the answer using the rule remembering the pattern in the rule.
- example: using both Trapezoidal Rule and Simpson's Rule, approximate the area under the curve of
 where 
- Trapezoidal Rule
- Simpson's Rule
- In comparing the results, they are different only by 0.01. They are both not exact answers but are very close approximations.
U Substitution
- In many cases, taking the antiderivative is not as easy as just finding the pattern or rule. In these cases, a method of substitution (called u substitution) has to be used. With u substitution, the variable u is substituted into the equation making it easier to solve the integral. In the following case, it does not look like it is possible to solve, but using substitution, it is very easy.
- In that example, traditional methods cannot be used to integrate. Instead, recognizing the parts of the integral make it easy to solve. Making u equal
 and du equal the derivative of u ( ), integration is easy.
- example: using u substitution, evaluate
 
Integration Links
Visual Calculus: Antiderivatives / Indefinite Integration - antiderivates flash presentation
Visual Calculus Drill: Substitutions  - u substitution drill with answer explanations
Antiderivatives - antidifferentiation site with anything and everything about antidifferentiation
Calculus-Help: Problem of the Week #14 - basic, yet challenging, integration problems
Visual Calculus: Fundamental Theorem of Calculus - Fundamental Theorem of Calculus flash presentations
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