Julie's copy of Volume of revolution 1

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Julie Crowley

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Julie Crowley 8 years, 6 months ago

Published this.

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Julie Crowley 11 years, 3 months ago

Created this as a copy of Volume of revolution 1.

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

There are 6 other versions that do you not have access to.

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	7
c	integer	5
b	integer	3
tv	number	931 pi^2
v	number	9188.602
sb	integer	1
sa	integer	1

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Name

Data type

Value

xxxxxxxxxx
 
sa*random(1..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`

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This variable doesn't seem to be used anywhere.

Parts

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1. Gap Number entry
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1. Step Mathematical expression

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Find the volume of this object.

$V=\;\;$ Unnamed gap

Enter your answer to 3 decimal places.

Click on Show steps for information on volumes of revolution. You will not lose any marks.

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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This question is used in the following exams:

Bridging Week2 Day 3 by Julie Crowley in Waste.
Methods of Integration by Deirdre Casey in Deirdre's workspace.
Methods of integration by Simon Thomas in Maths support.
Markus's copy of Methods of Integration by Markus Schmied in Markus's workspace.

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Taxonomy: Kind of activity

Taxonomy: Context

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History

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Checkpoint description

Julie Crowley 8 years, 6 months ago

Julie Crowley 11 years, 3 months ago

Error in variable testing condition

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Generated value: `integer`

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Julie's copy of Volume of revolution 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Julie Crowley 8 years, 6 months ago

Julie Crowley 11 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`