7.3 Numerical Integration

Numerical Integration

There are two new ways to find the area of a function using reimann sums. Instead of using small rectangles to find the area we are going to use trapezoids and parabolas.

TRAPAZOIDAL RULE
The equation we use to find the area under a function to the x-axis by using trapazoids is...



The first equation shows us the equation of a trapazoid. The second one shows us how to encorporate the equation for the area of a trapazoid to fit the function. As you can see, there is a pattern. You must multiply the first and last y value by one and the rest of the y values by 2. In the third equation you learn that the change in x is the same as a-b over the height.

SIMPSON'S RULE
The equation we use to find the area under a function to the x-axis by using parabolas is...



As you have noticed this equation also has a pattern like the trapazoidal equation. The first and last y values are multiplied by 1 and between these we alternate multiplying by 4 then 2. Make sure you end by multiplying by 4 before the last y value, otherwise the equation will not be accurate.

EXAMPLE PROBLEM
Calculate the area under the curve to 3 decimal places using the trapezoidal rule to approximate the given integral with the specified value of n.



(Trapazoidal Rule)

First we figure out that there are 4 intervals going from 0 to 2. This means that our f(x)'s will be in incriments for .5.
Second we find the y values for these incriments to 2.

x - y
0 - 1.000
.5 - 1.031
1 - 1.414
1.5 - 2.462
2 - 4.123

Now, we plug in these values for the equation, and then solve. :)






KATELYN UR UP NEXTTTTT!!!!!!!!! :) <3
i apologize that this blogg took so long. sorry.

http://mathworld.wolfram.com/NumericalIntegration.html
http://people.hofstra.edu/faculty/Stefan_waner/RealWorld/integral/numint.html



HAHAHAHAHA... i'm sorry i died laughing when i saw this. HAHAHAHA FABULOUS FIGURE!! :)