Double integral - limit is a polynomial - Numbas at mathcentre.ac.uk

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Bill Foster 6 years, 6 months ago

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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

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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	-1
b	integer	-1
d	integer	4
g	integer	2
f	integer	-2
ans1	number	-6
ans2	number	-69
h1	decimal	dec("1")
ans3	number	108
m	integer	4
n	integer	2
p1	integer	2
b1	integer	-2
c2	integer	2
c1	integer	4
d1	integer	3
d2	integer	-1
con	integer	3

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ans1

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

$I=$ Unnamed gap

Advice

a)

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

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)

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

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:

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

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Double integral - limit is a polynomial

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Bill Foster 6 years, 6 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

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Generated value: `integer`

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Double integral - limit is a polynomial

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 6 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

Generated value: `integer`