Changes

==Definite Integrals==

==Definite Integrals==

A definite integral is the area under the curve between two points on the function. In the picture below, the yellow area is "positive" and the blue area is "negative". The integral is evaluated by adding the positive area together and subtracting the negative area.

A definite integral is the area under the curve between two points on the function. In the picture below, the yellow area is "positive" and the blue area is "negative". The integral is evaluated by adding the positive area together and subtracting the negative area.

If the function f(x) is real rather than complex, then the definite integral is also known as a Riemann integral.

If the function f(x) is real rather than complex, then the definite integral is also known as a Riemann integral.

== Solving Definite Integrals ==

== Solving Definite Integrals ==

Solving a definite integral usually has two main steps: [[integration]] and [[subtraction]].

Solving a definite integral usually has two main steps: [[integration]] and [[subtraction]].

== Example 1==

== Example 1==

: <math>F(5) = {1 \over 3}5^3 = {125 \over 3}</math>

: <math>F(5) = {1 \over 3}5^3 = {125 \over 3}</math>

: <math>F(-3) = {1 \over 3}(-3)^3 = {-27 \over 3}</math>

: <math>F(-3) = {1 \over 3}(-3)^3 = {-27 \over 3}</math>

: <math>{125 \over 3} - {-27 \over 3} = {125 \over 3} + {27 \over 3} </math>

: <math>{125 \over 3} - {-27 \over 3} = {125 \over 3} + {27 \over 3} </math>

: <math>= {152 \over 3}</math>

: <math>= {152 \over 3}</math>

: <math>\int_4^{12}\frac{3x+11}{x^2-x-6}dx </math>

: <math>\int_4^{12}\frac{3x+11}{x^2-x-6}dx </math>

=== Integration ===

=== Integration ===

<br />See the [[Partial fractions in integration]] page for how to integrate.

<br />See the [[Partial fractions in integration]] page for how to integrate the above expression (it is the example).

<br />As shown on the page mentioned above:

<br />As shown on the page mentioned above:

: <math>\int\frac{3x+11}{x^2-x-6}dx=4ln|x-3|-ln|x+2|+c</math>

: <math>\int\frac{3x+11}{x^2-x-6}dx=4ln|x-3|-ln|x+2|+c</math>