1. Simple Polynomial

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Question statement

Evaluate the integral

$\int_{\var{b}}^{\var{d}} \int_{\var{a}}^{\var{c}} \var{k} \simplify{x^{n}} \simplify{y^{m}} \, \mathrm{d}x \, \mathrm{d}y \,.$

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			Name	Type	Generated Value

a	integer	1
b	integer	1
a2	integer	2
b2	integer	1
c	integer	3
d	integer	2
k	integer	2
n	integer	1
m	integer	2

			Name	Type	Generated Value

k1	rational	1
k2	number	8
k3	number	8
k4	number	2.6666666667
k5	number	7

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(0..(3-a2))

JME function reference

Description

start of x range {-5-c,5-c}

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

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Generated value: `integer`

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Statement
Advice

Advice

We can evaluate this iterated integral by straight-forward integration.

First integrate with respect to $x$ while treating $\simplify{y^{m}}$ as a constant.

$\begin{align} \int_{\var{b}}^{\var{d}}\int_{\var{a}}^{\var{c}}\var{k}\simplify{x^{n}}\simplify{y^{m}} \,\mathrm{d}x\,\mathrm{d}y &=\int_{\var{b}}^{\var{d}} \, \var{k}\frac{1}{\var{n+1}}x^{\var{n+1}}\bigg|_{x=\var{a}}^{x=\var{c}} \, \simplify{y^{m}} \,\mathrm{d}y \\ &=\int_{\var{b}}^{\var{d}}\simplify[fractionNumbers]{{k1}}(\var{c}^{\var{n+1}}-\var{a}^{\var{n+1}})\simplify{y^{m}} \,\mathrm{d}y \\ &=\int_{\var{b}}^{\var{d}}\simplify[fractionNumbers]{{k1}}\cdot \simplify{{k2}y^{m}} \,\mathrm{d}y \end{align}$
Now we can just integrate with respect to

$y$ as usual.

$\begin{align} \int_{\var{b}}^{\var{d}}\simplify[fractionNumbers]{{k3}}\simplify{y^{m}} \,\mathrm{d}y &= \simplify[fractionNumbers]{{k3}}\frac{1}{\var{m+1}}y^{\var{m+1}}\bigg|_{y=\var{b}}^{y=\var{d}} = \simplify[fractionNumbers]{{k4}}(\var{d}^{\var{m+1}}-\var{b}^{\var{m+1}}) = \simplify[fractionNumbers]{{k4}}\times\var{k5} = \simplify[fractionNumbers]{{k4*k5}} \end{align}$

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This question is used in the following exam:

ExamExample by Cristina Zambon in Cristina's workspace.

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Nicholas Barker 7 years ago

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1. Simple Polynomial

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Nicholas Barker 7 years ago

Nicholas Barker 7 years ago

Nicholas Barker 7 years, 3 months ago

Nicholas Barker 7 years, 3 months ago

Nicholas Barker 7 years, 4 months ago

Nicholas Barker 7 years, 4 months ago

Nicholas Barker 7 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Number entry

Advanced settings

Generated value: `integer`