Mid-Ordinate Rule

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Simon Thomas

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Simon Thomas 5 years, 2 months ago

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Simon Thomas 5 years, 2 months ago

Created this as a copy of Trapezium Rule 2.

Name	Status	Author	Last Modified
Trapezium Rule	draft	Simon Thomas	17/02/2020 10:26
Trapezium Rule 2	draft	Simon Thomas	17/02/2020 10:44
Mid-Ordinate Rule	draft	Simon Thomas	17/02/2020 10:50

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Question statement

Using the mid-ordinate rule with {n} strips:

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			Name	Type	Generated Value

a	integer	2
h	number	0.5
n	integer	5
b	number	4.5
c	integer	2
x	list	List of 5 items
y	list	List of 5 items
answer	number	26.73966
exactanswer	decimal	dec("2.687368212754142178277467444533122701995e+1")

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The mid-ordinate rule states that $\int^b_aydx\approx h[(y_{0.5}+y_{1.5}+...+y_{n-0.5})]$ where $h=\frac{b-a}{n}$

In our example $a=\var{a}, b=\var{b}$ and $n=\var{n}$ is the number of strips.

Hence $h=\frac{\var{b}-\var{a}}{\var{n}}=\var{h}$

So $\int^\var{b}_\var{a}{\var{c}^x}dx\approx\var{h}[(y_{0.5}+y_{1.5}+y_{2.5}+y_{3.5}+y_{4.5}]$

Our values of $x_i$ are given by $a+h \times i=\var{a}+\var{h}\times i$, as we vary $i$ from $0.5$ to $\var{n-0.5}$ with step-size 1

Thus we obtain $[x_{0.5},...,x_\var{n-0.5}] = \var{x}$

Now we can calculate each $y_i$ as $y_i = y(x_i) = {\var{c}^{x_i}}$

Thus we obtain $[y_{0.5},...,y_\var{n-0.5}] = \var{y}$

Hence $\int^\var{b}_\var{a}{\var{c}^x}dx\approx \var{h}[\var{y[0]}+\var{y[1]}+\var{y[2]}+\var{y[3]}+\var{y[4]}] \approx \var{answer}$

For comparison, the exact integral $\int\var{c}^x$ is given by $\frac{\var{c}^x}{\ln{\var{c}}}$. Hence the exact value of $\int^\var{b}_\var{a}{\var{c}^x}dx$ is $\frac{\var{c}^\var{b}}{\ln{\var{c}}}-\frac{\var{c}^\var{a}}{\ln{\var{c}}} = \var{precround(exactanswer,5)}$

The mid-ordinate rule approximation to the area under the curve can be seen below. The convexity (increasing gradient) of the curve explains why in this case the mid-ordinate rule gives an underestimate of the true integral.

{plotf(c,n,a,b)}

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This question is used in the following exam:

Numerical integration by Simon Thomas in Maths support.

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Simon Thomas 5 years, 2 months ago

Simon Thomas 5 years, 2 months ago

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Mid-Ordinate Rule

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 5 years, 2 months ago

Simon Thomas 5 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Number entry

Advanced settings

Generated value: `integer`