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Here we want to study the various methods from section 8.7 of the text

used to approximate the definite integral

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To begin with, we are going to concentrate on the example

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with 4 partitions. Thus 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 and 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.

To get the relevant Maple programs to work, we need to load a library.

Now we look at the right and left endpoint method.

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

Method Approx ln(2) error Left 0.759524 0.693147 0.066377 Right 0.634524 0.693147 0.058623 Midpoint 0.691220 0.693147 0.001927

Now we have a look at the Trapezoidal method and Simpson's method.

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Now we should think about the mistakes we make when we use approximations. First the rather mundane

right endpoint method.

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

Method = right, n = 4, R=0.634524 with Error = 0.058623 Method = right, n = 8, R=0.662872 with Error = 0.030275 Method = right, n = 16, R=0.677766 with Error = 0.015381 Method = right, n = 32, R=0.685396 with Error = 0.007751

Method = right, n = 64, R=0.689256 with Error = 0.003891 Method = right, n = 128, R=0.691198 with Error = 0.001949 Method = right, n = 256, R=0.692172 with Error = 0.000976 Method = right, n = 512, R=0.692659 with Error = 0.000488 Method = right, n = 1024, R=0.692903 with Error = 0.000244

Now we look at the Trapezoid method:

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Method = right, n = 4, T=0.697024 with Error = 0.003877 Method = right, n = 8, T=0.694122 with Error = 0.000975

Method = right, n = 16, T=0.693391 with Error = 0.000244 Method = right, n = 32, T=0.693208 with Error = 0.000061 Method = right, n = 64, T=0.693162 with Error = 0.000015 Method = right, n = 128, T=0.693151 with Error = 0.000004 Method = right, n = 256, T=0.693148 with Error = 0.000001

And finally Simpson's:

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

Method = right, n = 4, S=0.693254 with Error = 0.00010679 Method = right, n = 8, S=0.693155 with Error = 0.00000735 Method = right, n = 16, S=0.693148 with Error = 0.00000047 Method = right, n = 32, S=0.693147 with Error = 0.00000003