Wednesday, February 21, 2007

7.7 Approximate Integration

Hey everyone, ready for some fun math? K, well remember Reimann sums? it's backkkkk. but this time it's even more exciting because we use TRAPEZOIDS instead of RECTANGLES!

Let's take a look at a graph:

As you can see, the blue shaded regions are trapezoids, NOT rectangles. Therefore when we find the area, we will need to use the area of the trapezoids. In this example, there are five trapezoids. Their heights are B1, B2, B3, B4, B5 and B6. So now that we know the height, the width and the number of trapzeoids, we can find the total area.

Just in case we forgot geometry, the area of a trapzoid is:

* in this case, our "a" and "b" are B1, B2, B3...and so forth*

Finally we get the general equation which is called the trapezoid rule!

The trapezoid rule is just another way to approximate the area under a curve, but it is more accurate than using midpoints or Reimann sums.

But wait...don't you hate having to add numbers and then mulptify them? Don't you wish there was a simpler way? Well Simpson has the answer! Then using the same graph as above, the Simpson rule states:

*** Don't forget that the Simpson's Rule ONLY applies to even number of intervals.***

Example numero uno:
Use the trapezoid rule and the Simpson's rule to approximate: and also, n=4.

When n=4, Δx = π/4. Therefe when we use the trapezoid rule:

But wait! n=4, which is an even number! that means we can use the Simpson rule!

yay! now we are all gonna ace the quiz on friday right? of courseeee. but if you need more help, i found some really great sites:

  1. Unfortunately we couldn't see the visuals that Mr. French was going to show us, but this site has animations on creating trapezoids to approximate the area undert he curve.

  2. I don't know who the guy who wrote these Calc notes are, but he is AMAZING! This site offers detailed directions and explanations on how to use the two equations!




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