Volume of revolution 1

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Bill Foster

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Bill Foster 55 years, 2 months ago

Created this.

Name	Status	Author	Last Modified
Volume of revolution 1	draft	Bill Foster	06/08/2020 08:41
Julie's copy of Volume of revolution 1	draft	Julie Crowley	08/10/2016 21:59
Anna's copy of Julie's copy of Volume of revolution 1	draft	Anna Strzelecka	16/07/2019 17:52

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Question statement

Consider the solid object that is obtained when the function:

$y=\simplify[std]{{a}(cos(x)+{b})}$ is rotated by

$2\pi$ radians about the

$x$ -axis between the limits

$x=\var{c}\pi$ and

$x=\var{c+2}\pi$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-9
c	integer	2
b	integer	9
tv	number	130308.3869075828
v	number	130308.387
sb	integer	1
sa	integer	-1

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Data type

Value

xxxxxxxxxx
 
sa*random(2..9)

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Description

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

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Generated value: `integer`

-9

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Find the volume of this object.

$V=\;\;$ Unnamed gap

Enter your answer to 3 decimal places.

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Advice

Recall that if $V$ is the volume generated between the limits $x=a$ and $x=b$ by rotating the function about the $x$ -axis then $\displaystyle V=\pi\int_a^by^2\;dx$ .

So we have:

$\begin{eqnarray*} V&=&\pi\int_{\var{c}\pi}^{\var{c+2}\pi}\simplify[std]{{a^2}(cos(x)+{b})^2}\;dx\\ &=&\var{a^2}\pi\int_{\var{c}\pi}^{\var{c+2}\pi}\simplify[std]{cos(x)^2+{2*b}*cos(x)+{b^2}}\;dx\\ &=&\var{a^2}\pi\left[\simplify[std]{((1 / 4) Sin(2*x) + (1 / 2) * x + {2 * b} * Sin(x) + {b ^ 2} * x)}\right]_{\var{c}\pi}^{\var{c+2}\pi}\\ \end{eqnarray*}$
Here we have used the identity

$\cos(x)^2=\frac{1}{2}(1+\cos(2x))$ in order to integrate

$\cos(x)^2$ .

Since $\sin(n\pi)=0$ for all integers $n$ we see that:

$\begin{eqnarray*} V&=&\var{a^2}\pi\frac{\var{1+2b^2}}{\var{2}}\left(\var{c+2}\pi-\var{c}\pi\right)\\ &=&\var{a^2*(1+2b^2)}\pi^2\\ &=&\var{V}\mbox{ to 3 decimal places} \end{eqnarray*}$

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Taxonomy: mathcentre

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Bill Foster 55 years, 2 months ago

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Generated value: `integer`

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Volume of revolution 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`