Changes

An '''integral''' is a mathematical construction used in [[calculus]] to represent the area of a region in a plane bounded by the graph of a [[function]] in one [[real]] variable. [[Definite integral|Definite integrrals]] use the following notation:  

An '''integral''' is a mathematical construction used in [[calculus]] to represent the area of a region in a plane bounded by the graph of a [[function]] in one [[real]] variable. [[Definite integral|Definite integrrals]] use the following notation:  

<math>\int_a^b f(x)dx</math>

:<math>\int_a^b f(x)dx</math>

where ''a'' and ''b'' represent the lower and upper bounds of the [[interval]] being integrated over, ''f(x)'' represents the function being integrated (the '''integrand'''), and ''dx'' represents a dummy variable given various definitions, depending on the context of the integral.

where ''a'' and ''b'' represent the lower and upper bounds of the [[interval]] being integrated over, ''f(x)'' represents the function being integrated (the '''integrand'''), and ''dx'' represents a dummy variable given various definitions, depending on the context of the integral.

<h3>Additive Distribution</h3>

<h3>Additive Distribution</h3>

If an integral contains two functions added together, it may be rewritten as two separate integrals, each containing one function.

If an integral contains two functions added together, it may be rewritten as two separate integrals, each containing one function.

*<math>\int_{a}^{b}(f(x)+g(x))dx=\int_{a}^{b}f(x)dx+\int_{a}^{b}g(x)dx</math>

:<math>\int_{a}^{b}(f(x)+g(x))dx=\int_{a}^{b}f(x)dx+\int_{a}^{b}g(x)dx</math>

<h3>Constant Multiplicative Distribution</h3>

<h3>Constant Multiplicative Distribution</h3>

If an integral contains a function multiplied by a constant, it may be rewritten as the constant times the integral of the function.

If an integral contains a function multiplied by a constant, it may be rewritten as the constant times the integral of the function.

*<math>\int_{a}^{b}cf(x)dx=c\int_{a}^{b}f(x)dx</math>

:<math>\int_{a}^{b}cf(x)dx=c\int_{a}^{b}f(x)dx</math>

<h3>Area</h3>

<h3>Area</h3>

Any integral that contains the same upper and lower bounds is equal to zero.

Any integral that contains the same upper and lower bounds is equal to zero.

*<math>\int_{a}^{a}f(x)dx=0</math>  

:<math>\int_{a}^{a}f(x)dx=0</math>  

<h3>Separation</h3>

<h3>Separation</h3>

An integral may be rewritten as the sum of two integrals with adjacent bounds.

An integral may be rewritten as the sum of two integrals with adjacent bounds.

*<math>\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx</math>, <math>c\in(a,b)</math>

:<math>\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx</math>, <math>c\in(a,b)</math>

<h3>Additive Inversion</h3>

<h3>Additive Inversion</h3>

Any integral is equal to the additive inverse of the same integral with reversed upper and lower bounds.

Any integral is equal to the additive inverse of the same integral with reversed upper and lower bounds.

*<math>\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx</math>

:<math>\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx</math>

==Antiderivative vs Integration==

==Antiderivative vs Integration==

There are important differences between the anti-derivative and integration. An anti-derivative of a function <math>f(x)</math> is a function <math>g(x)</math> such that,

There are important differences between the anti-derivative and integration. An anti-derivative of a function <math>f(x)</math> is a function <math>g(x)</math> such that,

<math>\frac{d}{dx}g(x)=f(x)</math>

:<math>\frac{d}{dx}g(x)=f(x)</math>

The integral of a function can be evaluated using its antiderivative,

The integral of a function can be evaluated using its antiderivative,

<math>\int_a^b f(x)dx=g(b)-g(a)</math>

:<math>\int_a^b f(x)dx=g(b)-g(a)</math>

This works for the kind of functions encountered in late high school and early university mathematics. It is, however, an incomplete method. For example one cannot write the anti-derivative of <math>e^{x^{2}}</math> in terms of familiar functions (such as [[trigonometric function]]s, [[exponential]]s, and [[logarithm]]s) and function operations.

This works for the kind of functions encountered in late high school and early university mathematics. It is, however, an incomplete method. For example one cannot write the anti-derivative of <math>e^{x^{2}}</math> in terms of familiar functions (such as [[trigonometric function]]s, [[exponential]]s, and [[logarithm]]s) and function operations.