6. Step Function

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Question statement

The function $g(x,y)$ is a step function taking the following values on each of the rectangles shown below.

{domain()}

Calculate the integral

$\iint_D g(x,y) dx dy$

of $g$ over the whole rectangle $D=[0,{\var{MX()}}]\times[0,{\var{N[y]}}]$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

sm	range	2..5
sn	range	1..5
m1	integer	4
m2	integer	8
m3	integer	10
m4	integer	13
n1	integer	3
n2	integer	8
n3	integer	9
n4	integer	13

			Name	Type	Generated Value

x	integer	2
y	integer	4
M	vector	vector(0,4,8,10,13)
N	vector	vector(0,3,8,9,13)
p	range	0..9
Q	matrix	Matrix of size 5×5

Automatically regenerate variables?

Name

Data type

Value

Numbers between and (inclusive) with step size

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `range`

2..5

This variable doesn't seem to be used anywhere.

Advice

To calculate the integral of the first rectangle $[0, \var{M[1]}]\times[0, \var{N[1]}]$ we just need to find its area and multiply by the value of $g$ inside it, because $g$ is constant.

\[ I_{1,1} = \int_0^{ \var{N[1]}}\int_0^{ \var{M[1]}} g(x,y)\, \mathrm{d}x\,\mathrm{d}y = \var{M[1]} \cdot \var{N[1]} \cdot \var{Q[1][1]} = \var{M[1]*N[1]*Q[1][1]} \]

To integrate over the whole grid we integrate over every other rectangle and add all the results together.

\begin{align}
I =& I_{1,1} &&+ I_{1,2} &&+... \\
& \int_0^{\var{N[1]}}\int_0^{\var{M[1]}} g(x,y)\, \mathrm{d}x\,\mathrm{d}y &&+ \int_0^{\var{N[1]}}\int_{\var{M[1]}}^{\var{M[2]}} g(x,y)\, \mathrm{d}x\,\mathrm{d}y &&+ ... \\
=& \var{M[1]} \cdot \var{N[1]} \cdot \var{Q[1][1]} &&+ ( \var{M[2]}-\var{M[1]}) \cdot \var{N[1]} \cdot \var{Q[1][2]} &&+ ... \\
=& \var{M[1]*N[1]*Q[1][1]} &&+ \simplify{{(M[2]-M[1])*N[1]*Q[1][2]}} &&+ ... \\
=& {\var{ans()}}
\end{align}

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This question is used in the following exams:

Test exam by Tyson Ritter in Tyson's workspace.
ExamExample by Cristina Zambon in Cristina's workspace.
analysis by Daniel Atcheson in Daniel's workspace.

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Nicholas Barker 7 years, 1 month ago

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Nicholas Barker 7 years, 2 months ago

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6. Step Function

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Nicholas Barker 7 years, 1 month ago

Nicholas Barker 7 years, 1 month ago

Nicholas Barker 7 years, 2 months ago

Error in variable testing condition

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Warning

Generated value: range

Parts

Number entry

Advanced settings

Generated value: `range`