Friday, April 27, 2007

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.

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.

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.

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. :)

i apologize that this blogg took so long. sorry.

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



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