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

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

Calculate the following repeated integrals.

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			Name	Type	Generated Value

a	integer	3
p2	integer	2
c	integer	-4
b	integer	-2
d	integer	2
g	integer	5
f	integer	-1
ans1	number	-72
ans2	number	211
h1	decimal	dec("1")
ans3	number	186
m	integer	4
n	integer	2
p1	integer	1
b1	integer	3
c2	integer	2
c1	integer	2
d1	integer	2
d2	integer	-1
con	integer	15

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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

Answer must be an integer.

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

Calculating the inner integral we have:

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

The outer integral gives:

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

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

Calculating the inner integral we have :

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

Finally the outer integral gives:

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

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

Calculating the inner integral we have :

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

Finally the outer integral gives:

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

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Double integral - limit is a polynomial,

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Taxonomy: mathcentre

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Taxonomy: Context

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: integer

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Generated value: `integer`