Julie's copy of Definite integration 1

Error

There was an error loading the page.

Contributors

Julie Crowley

History

○

Julie Crowley 8 years, 5 months ago

Published this.

○

Julie Crowley 11 years, 2 months ago

Created this as a copy of Definite integration 1.

Name	Status	Author	Last Modified
Definite integration 1	draft	Bill Foster	06/08/2020 08:41
Julie's copy of Definite integration 1	draft	Julie Crowley	08/10/2016 20:51
cormac's copy of Julie's copy of Definite integration 1	draft	cormac breen	23/02/2017 18:19
Simon's copy of Julie's copy of Definite integration 1	draft	Simon Thomas	21/03/2019 09:27
Definite integration 1	draft	Jo-Ann Lyons	24/04/2019 15:47
Danny's copy of Definite integration 1	draft	Danny Tran	21/05/2019 05:12

There are 17 other versions that do you not have access to.

Question statement

Evaluate the following definite integral.

Give any introductory information the student needs.

			Name	Type	Generated Value

ans1	number	2847.90
ans2	decimal	dec("1.10")
ans3	number	-0.08
ans4	number	0.1606
b4	integer	1
b1	integer	4
b2	integer	2
d1	integer	-2
s2	integer	1
s7	integer	1
s6	integer	-1
m4	integer	2
m3	integer	5
m2	integer	1
tol	number	0.01
a1	integer	3
tans4	number	0.1606027941
c1	integer	3
tans1	number	2847.9
tans3	number	-0.08
tol1	number	0.0001
p	integer	-1
t	integer	1
w	integer	0
n4	integer	-1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
precround(tans1,2)

JME function reference

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: `number`

2847.90

← Depends on:

tans1

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Number entry
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$I=\int_1^{\var{b1}}\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\;dx$

$I=\;\;$ Unnamed gap

Input your answer to 2 decimal places.

Advice

$I=\int_1^\var{b1}\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\;dx$
Expand the parentheses to obtain:

$\begin{eqnarray*}I &=& \int_1^\var{b1} \simplify[std]{{a1 ^ 2} * x ^ 4 + {2 * a1 * c1} * x ^ 3+ {c1 ^ 2+2*a1*d1} * x ^ 2 + {2 * c1 * d1} * x+ {d1 ^ 2} }\;dx\\ &=&\left[\simplify[std]{{a1 ^ 2}/5 * x ^ 5 + {2 * a1 * c1}/4 * x ^ 4+ {c1 ^ 2+2*a1*d1}/3 * x ^ 3 + {2 * c1 * d1}/2 * x^2+ {d1 ^ 2}x }\right]_1^\var{b1}\\ &=&\var{tans1}\\ \\&=&\var{ans1}\mbox{ to 2 decimal places} \end{eqnarray*}$

$\begin{eqnarray*}I&=&\int_0^{\var{b2}}\simplify[std]{1/(x+{m2})}\;dx\\ &=&\left[\ln(x+\var{m2})\right]_0^{\var{b2}}\\ &=& \ln(\var{b2+m2})-\ln(\var{m2})\\ &=&\ln\left(\frac{\var{b2+m2}}{\var{m2}}\right)\\ &=&\var{ans2}\mbox{ to 2 decimal places} \end{eqnarray*}$

$I=\int_0^\pi\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\;dx$
We use integration by parts.

Recall that:

$\int u\frac{dv}{dx}\;dx=uv-\int \frac{du}{dx}\;v\;dx$
Here we set

$u=x$ and

$\displaystyle \frac{dv}{dx}=\simplify[std]{ {w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x)}$

Hence

$v=\simplify[std]{({-w}/ {m3}) * Cos({m3} * x) + {1 -w} * (({1-w}/ {m3}) * Sin({m3} * x))}$

$\begin{eqnarray*} I&=&\left[\simplify[std]{{-w}*((x / {m3}) * Cos({m3} * x)) + {1 -w} * ((x / {m3}) * Sin({m3} * x))}\right]_0^\pi -\int_0^\pi\simplify[std]{ ({ -w} / {m3} )* Cos({m3} * x) + {1 -w} * (1 / {m3} * Sin({m3} * x))}\;dx\\ &=&\simplify[std]{({-w*cos(m3*pi)})*({pi}/{m3})}-\left[\simplify[std]{{ -w} * (1 / {m3 ^ 2})* Sin({m3} * x) -({1 -w} * (1 / {m3 ^ 2}) * Cos({m3} * x))}\right]_0^\pi\\ &=& \var{ans3}\mbox{ to 2 decimal places} \end{eqnarray*}$
d)

$I=\int_0^{\var{b4}}\simplify[std]{x ^ {m4} * Exp({n4} * x)}\;dx$

Use integration by parts twice with $u=x^2$ and $\displaystyle \frac{dv}{dx}=\simplify[std]{e^({n4}x)}\Rightarrow v = \simplify[std]{1/{n4}e^({n4}x)}$

$\begin{eqnarray*} I&=&\left[\simplify[std]{x^2/{n4}Exp({n4} * x)}\right]_0^{\var{b4}}+\simplify[std]{2/{abs(n4)}DefInt(x*Exp({n4} * x),x,0,{b4})}\\ &=&\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\left(\left[\simplify[std]{x/{n4}*e^({n4}*x)}\right]_0^{\var{b4}}+\simplify[std]{1/{abs(n4)}DefInt(e^({n4}x),x,0,{b4})}\right)\\ &=&\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\left(\simplify[std]{{b4}/{n4}*e^{p}-1/{n4}}\left[\simplify[std]{1/{n4}*e^({n4}*x)}\right]_0^{\var{b4}}\right)\\ &=&\simplify[std]{({b4 ^ 2} / {n4}) * Exp({p}) -(({2 * b4} / {n4 ^ 2}) * Exp({p})) + (2 / {n4 ^ 3}) * (Exp({p}) -1)}\\ &=&\var{ans4}\mbox{ to 4 decimal places} \end{eqnarray*}$

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet

This question is used in the following exams:

Wk2 Day3 by Julie Crowley in Waste.
Bridging Calculus Review by Julie Crowley in Waste.
Bridging Week 2 Day 1 by Julie Crowley in Waste.
Integration by Julie Crowley in Julie's workspace.
All of Integration by Julie Crowley in Julie's workspace.
Integration Rules by Julie Crowley in Julie's workspace.
Bridging Calculus Review by Marlon Arcila in Tst.
Introduction to Integration by Deirdre Casey in Deirdre's workspace.
Introduction to Integration by Violeta CIT in MATH6006 Integration.
Nick's copy of Introduction to Integration by Nick Walker in Nick's workspace.
Nick's copy of Introduction to Integration by Nick Walker in Nick's workspace.
Nick's copy of Introduction to Integration by Nick Walker in Nick's workspace.
NUMBAS-Definite Integration by Katy Dobson in Katy's workspace.
Johnny's copy of Integration by Johnny Yi in Diagnostic Test Materials.
Max's copy of Integration by Max S in Max's workspace.
Integration by Julie Crowley in MATH6002 Calculus and Statistics for the Biological Sciences.
Ann's copy of Integration by Ann Smith in Ann's workspace.
Maria's copy of Introduction to Integration by Maria Aneiros in Maria's workspace.
Ann's copy of Ann's copy of Integration by Ann Smith in Year One Engineering Maths.
Maths 2 Quiz 4 Integration Basic & Substitution by Aoife O'Brien in Aoife's workspace.
Introduction to Integration by Violeta CIT in Bridging 2020.
Blathnaid's copy of Ann's copy of Integration by Blathnaid Sheridan in Blathnaid's workspace.
Stefania's copy of Maria's copy of Introduction to Integration by Stefania Sitzia in Week 07.
Introduction to Integration for CA 2 by Violeta CIT in Bridging 2020.
Michael's copy of Introduction to Integration by Michael Proudman in .Integral Calculus.
Integration by Julie Crowley in Get Maths Fit Workshops.
Introduction to Integration by Gagan Aggarwal in Gagan's workspace.
Sara's copy of Integration by Sara Santos in Sara's workspace.
Introduction to Integration by Anna Cartlidge in Glasgow Numbas Question Pool.
Michael's copy of Ann's copy of Integration by Michael Barkley in 01-UoL-STEM-FY-FS0032-OLD.

Loading...

Error

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Julie Crowley 8 years, 5 months ago

Julie Crowley 11 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: `number`

Parts

Gap-fill

Loading...

Error

Julie's copy of Definite integration 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Julie Crowley 8 years, 5 months ago

Julie Crowley 11 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`