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Trapezoidal & Simpson's Rule

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Focus on two functions and approximate them using the Trapezoidal Rule and Simpson's Rule:

Given the following function:

f(x)= -2?3+?(16-?(x-2)?^2 )

Plot your function from x=0 to x=5.

You can plot this function using a graphing calculator. Using the TI83/84, click on y = and type in the function. Then click on graph to get the following graph:

The plot area here is from -1 < x < 5 and from -1 < y < 1

B) Estimate the area under the curve for x=0 to x=4. Use any brute method (not
the approximation methods listed in your text).

We can estimate this area by drawing two triangles and two trapezoids in the region from x = 0 to x = 4. Like this:

The two triangles each have base of length 1. To find the height, we can evaluate the function at x = 1.
f(1) = -2?3 + ?(16 - (1 - 2)2) = -2?3 + ?(16 - 1) = -2?3 + ?15 ? .40888

So the area of each triangle is ½ (1)(.40888) = .20444

The two trapezoids have a height of 1, we know that one of the bases is .40888 and we can find the second base by evaluating the function at x = 2:
f(2) = -2?3 + ?(16 - (2 - 2)2) = -2?3 + ?(16) = -2?3 + 4 ? ...

Solution Summary

Trapezoidal and Simpson`s Rule are examined.

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