4. One Type Easier than the Other

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Question statement

Evaluate the integral

$\int_0^\simplify[fractionNumbers]{{b}} \int_\simplify{({a}x)^(1/{n})}^\var{c} \simplify{{k}e^({m}y^{n+1})} \, \mathrm{d}y \, \mathrm{d}x .$

{domain()}

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			Name	Type	Generated Value

k1	rational	1/8
num	number	1
den	number	8
gcd	number	2

			Name	Type	Generated Value

a	integer	2
b	rational	1
c	integer	2
k	integer	2
m	integer	2
n	integer	3

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Value

xxxxxxxxxx
 
k/(a*m*(n+1))

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Generated value: `rational`

1/8

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We cannot integrate $\simplify{{k}e^({m}y^{n+1})}$ with respect to $y$, so we must change type and integrate with respect to $x$.

\begin{align}
I=\int_0^\var{c} \int_0^{\simplify{y^{n}/{a}}} \var{k}e^\simplify{{m}y^{n+1}} \, \mathrm{d}x \, \mathrm{d}y
&=\int_0^\var{c} \var{k}xe^\simplify{{m}y^{n+1}} \bigg|_{x=0}^{x=\simplify{y^{n}/{a}}} \, \mathrm{d}y \\
&=\int_0^\var{c} \simplify{1/{a}y^{n}}\cdot\var{k}e^\simplify{{m}y^{n+1}} \, \mathrm{d}y \\
&=\int_0^\var{c} \simplify[fractionNumbers]{{k/a}}\simplify{y^{n}}e^\simplify{{m}y^{n+1}} \, \mathrm{d}y \\
\end{align}

\begin{align}
\mathrm{I}
&=\simplify{1/({m*(n+1)}y^{n})}\cdot\simplify[fractionNumbers]{{k/a}}\simplify{y^{n}}e^\simplify{{m}y^{n+1}} \bigg|_{y=0}^{y=\var{c}} \\
&=\simplify[fractionNumbers]{{k1}}e^\simplify{{m}y^{n+1}} \bigg|_{y=0}^{y=\var{c}} \\
&=\simplify[fractionNumbers]{{k1}}(\simplify{e^{m*c^(n+1)}}-1) \\
\end{align}

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Nicholas Barker 7 years ago

Nicholas Barker 7 years ago

Nicholas Barker 7 years, 3 months ago

Error in variable testing condition

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Generated value: `rational`

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4. One Type Easier than the Other

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

Error in variable testing condition

Error

Warning

Generated value: rational

Parts

Mathematical expression

Advanced settings

Checking accuracy

Variable value generators

Pattern restriction

Variables

String restrictions

Generated value: `rational`