Ugur's copy of Double integral - limit is a polynomial

Error

There was an error loading the page.

Contributors

Feedback

From users who are members of Ugur's workspace :

Ugur Efem

said

Ready to use

2 years ago

History

○

Ugur Efem 2 years ago

Published this.

○

Ugur Efem 2 years ago

Gave some feedback: Ready to use

○

Ugur Efem 2 years, 1 month ago

Created this as a copy of Double integral - limit is a polynomial.

Name	Status	Author	Last Modified
Double integral - limit is a polynomial	Ready to use	Christian Lawson-Perfect	20/11/2019 14:51
Luis's copy of Double integral - limit is a polynomial	draft	Luis Hernandez	30/11/2018 14:59
Ugur's copy of Double integral - limit is a polynomial	Ready to use	Ugur Efem	04/04/2023 15:35

There are 8 other versions that do you not have access to.

Question statement

Calculate the following repeated integrals.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	1
p2	integer	1
c	integer	-2
b	integer	-1
d	integer	6
g	integer	9
f	integer	3
ans1	number	204
ans2	number	66
h1	decimal	dec("-2")
ans3	number	-6
m	integer	5
n	integer	2
p1	integer	2
b1	integer	2
c2	integer	1
c1	integer	3
d1	integer	-6
d2	integer	1
con	integer	3

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(1..4)

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: `integer`

→ Used by:

ans1

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Number entry
  Add
Part b) Gap-fill
1. Gap Number entry
  Add
Part c) 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.

$I = \int_0^{\var{a}}\;\int_{\var{f}}^{\var{g}}\simplify[all]{({b}+{c}*x+{d}*y)}\;dy \;dx$

$I=$ Unnamed gap

Advice

a)

$I = \int_0^{\var{a}} \;\int_{\var{f}}^{\var{g}}\simplify[all]{({b}+{c}*x+{d}*y)} \; \mathrm{d}y\; \mathrm{d}x$

Calculating the inner integral, we have:

$\begin{align} \int_{\var{f}}^{\var{g}}\simplify[all,!noleadingminus,!collectNumbers]{({b}+{c}*x+{d}*y)} \; \mathrm{d}y &= \left[\simplify[all,!noleadingminus,!collectNumbers]{{b}y+{c}*x*y+{d}*y^2/2}\right]_{\var{f}}^{\var{g}}\\ &= \simplify[all,!noleadingminus,!collectNumbers]{{b} * {g} + {c} * {g} * x + {d / 2} * {g ^ 2} + {b} * { -f} + {c} * { -f} * x + {d / 2} * { -(f ^ 2)}} \\ &= \simplify[all,!noleadingminus,!collectNumbers]{ {b * g -(b * f) + (d / 2) * (g ^ 2 -(f ^ 2))} + {c * g -(c * f)} * x} \end{align}$

The outer integral gives:

$\begin{align} I &= \simplify[std]{DefInt({b * g -(b * f) + (d / 2) * (g ^ 2 -(f ^ 2))} + {c * g -(c * f)} * x,x,0,{a}) } \\ &= \left[\simplify[std]{{b * g -(b * f) + (d / 2) * (g ^ 2 -(f ^ 2))} * x + {(c * g -(c * f)) / 2} * x ^ 2}\right]_0^{\var{a}} \\ &= \var{ans1} \end{align}$

b)

$I=\var{(m + 1) * (m + n + 2)} \int_0^1 \; \mathrm{d}x \int_x^{\var{b1}}\simplify[std]{({n + 1} * x ^ {m} * y ^ {n})} \; \mathrm{d}y$

Calculating the inner integral, we have:

$\begin{align} \int_x^{\var{b1}}\simplify[std]{({n + 1} * x ^ {m} * y ^ {n})}dy &= \left[x^{\var{m}}y^{\var{n+1}}\right]_x^{\var{b1}} \\ &= \simplify{{b1 ^ (n + 1)}* x ^ {m} -(x ^ {m + n + 1})} \end{align}$

Finally the outer integral gives:

$\begin{align} I &= \var{(m + 1) * (m + n + 2)}\int_0^1\simplify[std]{{b1^ (n + 1)} * x ^ {m} -(x ^ {m + n + 1})} \; \mathrm{d}x \\ &= \simplify[std]{ {(m + 1) * (m + n + 2)} * ({b1 ^ (n + 1)} / {m + 1} -(1 / {m + n + 2})) } \\ &= \var{ans2} \end{align}$

c)

$I=\var{con}\int_{-1}^1 \; \int_0^{\simplify[all]{{c2}+{d2}*x^{p2}}}\simplify[all]{{c1}+{d1}*y^{p1}} \; \mathrm{d}y \; \mathrm{d}x$

Calculating the inner integral, we have:

$\begin{align} \int_0^{\simplify[all]{{c2}+{d2}*x^{p2}}}\simplify[all]{{c1}+{d1}*y^{p1}} \; \mathrm{d}y &= \left[\simplify[all]{{c1} * y + {d1 / (p1 + 1)} * y ^ {p1 + 1}}\right]_0^{\simplify[all]{{c2}+{d2}*x^{p2}}} \\ &= \simplify[std]{{c1} * ({c2} + {d2} * x ^ {p2}) + {d1 / (p1 + 1)} * ({c2} + {d2} * x ^ {p2}) ^ {p1 + 1}} \\ &= \simplify[std]{ {c1 * c2} + {c1 * d2} * x ^ {p2} + {p1 -1} * {h1} * ({c2 ^ 3} + {3 * c2 ^ 2 * d2} * x ^ {p2} + {3 * c2 * d2 ^ 2} * x ^ {2 * p2} + {d2} ^ 3 * x ^ {3 * p2}) + {2 -p1} * {h1} * ({c2 ^ 2} + {2 * c2 * d2} * x ^ {p2} + {d2 ^ 2} * x ^ {2 * p2})} \\ &= \simplify[std,collectNumbers]{{c1 * c2 + (p1 -1) * h1 * c2 ^ 3 + (2 -p1) * h1 * c2 ^ 2} + {c1 * d2 + (p1 -1) * h1 * 3 * c2 ^ 2 * d2 + (2 -p1) * h1 * 2 * c2 * d2} * x ^ {p2} + {(p1 -1) * h1 * 3 * c2 * d2 ^ 2 + (2 -p1) * h1 * d2 ^ 2} * x ^ {2 * p2} + {(p1 -1) * h1 * d2 ^ 3} * x ^ {3 * p2}} \end{align}$

Finally the outer integral gives:

$\begin{align} I &= \simplify[std]{{con} * DefInt({c1 * c2 + (p1 -1) * h1 * c2 ^ 3 + (2 -p1) * h1 * c2 ^ 2} + {c1 * d2 + (p1 -1) * h1 * 3 * c2 ^ 2 * d2 + (2 -p1) * h1 * 2 * c2 * d2} * x ^ {p2} + {(p1 -1) * h1 * 3 * c2 * d2 ^ 2 + (2 -p1) * h1 * d2 ^ 2} * x ^ {2 * p2} + {(p1 -1) * h1 * d2 ^ 3} * x ^ {3 * p2},x, -1,1)} \\ &= \var{ans3} \end{align}$

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
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet

This question is used in the following exam:

Double Integrals by Ugur Efem in Ugur's workspace.

Loading...

Error

Ugur's copy of Double integral - limit is a polynomial

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ugur Efem 2 years ago

Ugur Efem 2 years ago

Ugur Efem 2 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

Loading...

Error

Ugur's copy of Double integral - limit is a polynomial

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ugur Efem 2 years ago

Ugur Efem 2 years ago

Ugur Efem 2 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

Generated value: `integer`