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Calculus 3 : Multiple Integration

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Example Questions

Example Question #451 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3.5}^{0.5}\int_{-4.5}^{2.25}\int_{-3}^{1.5}(\frac{(2^y\cdot 3^{(\frac{z}{2})})}{2^x})dxdydz\end{align*}$

Possible Answers:

-39.93

159.73

-319.46

79.86

Correct answer:

159.73

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral, and keeping savvy about }\\&\text{utilizing integral rules:}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\end{align*}$

$\begin{align*}&\int_{-3.5}^{0.5}\int_{-4.5}^{2.25}\int_{-3}^{1.5}(\frac{(2^y\cdot 3^{(\frac{z}{2})})}{2^x})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3.5}^{0.5}\int_{-4.5}^{2.25}\int_{-3}^{1.5}(\frac{(2^y\cdot 3^{(\frac{z}{2})})}{2^x})dxdydz=\int_{-3.5}^{0.5}\int_{-4.5}^{2.25}(-\frac{(2^{(y - x)}\cdot 3^{(\frac{z}{2})})}{ln(2)})dydz|_{-3}^{1.5}\\&\int_{-3.5}^{0.5}\int_{-4.5}^{2.25}(-\frac{(2^y\cdot 3^{(\frac{z}{2})}\cdot (2^{(\frac{1}{2})} - 32))}{(4ln(2))})dydz=\int_{-3.5}^{0.5}(\frac{(2^y\cdot (8\cdot 3^{(\frac{z}{2})} -\frac{ (2^{(\frac{1}{2})}\cdot 3^{(\frac{z}{2})})}{4}))}{ln(2)^{2}})dz|_{-4.5}^{2.25}\\&\int_{-3.5}^{0.5}(-\frac{(3^{(\frac{z}{2})}\cdot (\frac{2^{(\frac{1}{2})}}{4}- 32\cdot 2^{(\frac{1}{4})} + 2^{(\frac{3}{4})} -\frac{ 1}{64}))}{ln(2)^{2}})dz=-\frac{(2\cdot 3^{(\frac{z}{2})}\cdot (\frac{2^{(\frac{1}{2})}}{4}- 32\cdot 2^{(\frac{1}{4})} + 2^{(\frac{3}{4})} -\frac{ 1}{64}))}{(ln(2)^{2}ln(3))}|_{-3.5}^{0.5}=159.73\end{align*}$

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Example Question #452 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{9.5}\int_{-3.5}^{1.5}\int_{10}^{14}(\frac{(17\cdot 2^{(\frac{y}{2})})}{(3xz^{2})})dxdydz\end{align*}$

Possible Answers:

$-1.17\cdot10^{-1}$

1.4 0

$4.68\cdot10^{-1}$

-2.81

Correct answer:

$4.68\cdot10^{-1}$

Explanation:

$\begin{align*}&\int_{6}^{9.5}\int_{-3.5}^{1.5}\int_{10}^{14}(\frac{(17\cdot 2^{(\frac{y}{2})})}{(3xz^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{9.5}\int_{-3.5}^{1.5}\int_{10}^{14}(\frac{(17\cdot 2^{(\frac{y}{2})})}{(3xz^{2})})dxdydz=\int_{6}^{9.5}\int_{-3.5}^{1.5}(\frac{(17\cdot 2^{(\frac{y}{2})}ln(x))}{(3z^{2})})dydz|_{10}^{14}\\&\int_{6}^{9.5}\int_{-3.5}^{1.5}(\frac{(17\cdot 2^{(\frac{y}{2})}ln(\frac{7}{5}))}{(3z^{2})})dydz=\int_{6}^{9.5}(\frac{(34\cdot 2^{(\frac{y}{2})}ln(\frac{7}{5}))}{(3z^{2}ln(2))})dz|_{-3.5}^{1.5}\\&\int_{6}^{9.5}(-\frac{(17ln(\frac{7}{5})\cdot (2^{(\frac{1}{4})} - 4\cdot 2^{(\frac{3}{4})}))}{(6z^{2}ln(2))})dz=\frac{(17\cdot 2^{(\frac{1}{4})}ln(\frac{7}{5}) - 68\cdot 8^{(\frac{1}{4})}ln(\frac{7}{5}))}{(6zln(2))}|_{6}^{9.5}=4.68\cdot10^{-1}\end{align*}$

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Example Question #453 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{3.5}^{8.5}\int_{-8}^{-3}\int_{-10}^{-7.5}(\frac{(3^zcos(4x)cos(4y))}{7})dxdydz\end{align*}$

Possible Answers:

173.44

57.81

-43.36

-346.87

Correct answer:

173.44

Explanation:

$\begin{align*}&\int_{3.5}^{8.5}\int_{-8}^{-3}\int_{-10}^{-7.5}(\frac{(3^zcos(4x)cos(4y))}{7})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3.5}^{8.5}\int_{-8}^{-3}\int_{-10}^{-7.5}(\frac{(3^zcos(4x)cos(4y))}{7})dxdydz=\int_{3.5}^{8.5}\int_{-8}^{-3}(\frac{(3^zcos(4y)sin(4x))}{28})dydz|_{-10}^{-7.5}\\&\int_{3.5}^{8.5}\int_{-8}^{-3}(-\frac{(3^zcos(4y)\cdot (\frac{sin(30)}{4}-\frac{ sin(40)}{4}))}{7})dydz=\int_{3.5}^{8.5}(-\frac{(3^zsin(4y)\cdot (sin(30) - sin(40)))}{112})dz|_{-8}^{-3}\\&\int_{3.5}^{8.5}(\frac{(3^z\cdot (sin(12) - sin(32))\cdot (sin(30) - sin(40)))}{112})dz=-\frac{(3^z\cdot (\frac{cos(2)}{2}-\frac{ cos(8)}{2}-\frac{ cos(18)}{2}+\frac{ cos(28)}{2}+\frac{ cos(42)}{2}-\frac{ cos(52)}{2}-\frac{ cos(62)}{2}+\frac{ cos(72)}{2}))}{(112ln(3))}|_{3.5}^{8.5}=173.44\end{align*}$

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Example Question #451 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{5.5}\int_{3}^{7.5}\int_{-4.5}^{-1}(8e^{(-2y)}e^{(-2z)}e^{(x)})dxdydz\end{align*}$

Possible Answers:

$-1.45\cdot10^{-6}$

$4.35\cdot10^{-6}$

$-1.31\cdot10^{-5}$

$1.45\cdot10^{-6}$

Correct answer:

$4.35\cdot10^{-6}$

Explanation:

$\begin{align*}&\int_{3}^{5.5}\int_{3}^{7.5}\int_{-4.5}^{-1}(8e^{(-2y)}e^{(-2z)}e^{(x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{5.5}\int_{3}^{7.5}\int_{-4.5}^{-1}(8e^{(-2y)}e^{(-2z)}e^{(x)})dxdydz=\int_{3}^{5.5}\int_{3}^{7.5}(8e^{(-2y)}e^{(-2z)}e^{(x)})dydz|_{-4.5}^{-1}\\&\int_{3}^{5.5}\int_{3}^{7.5}(e^{(- 2y - 2z)}\cdot (8e^{(-1)} - 8e^{(-\frac{9}{2})}))dydz=\int_{3}^{5.5}(-4e^{(-\frac{9}{2})}e^{(- 2y - 2z)}\cdot (e^{(\frac{7}{2})} - 1))dz|_{3}^{7.5}\\&\int_{3}^{5.5}(4e^{(-2z)}\cdot (e^{(-16)} - e^{(-\frac{39}{2})})\cdot (e^{(9)} - 1))dz=-2e^{(-2z)}\cdot (e^{(-16)} - e^{(-\frac{39}{2})})\cdot (e^{(9)} - 1)|_{3}^{5.5}=4.35\cdot10^{-6}\end{align*}$

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Example Question #321 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{8.5}\int_{-4.5}^{-2}\int_{3}^{5.5}(\frac{(53\cdot 2^ycos(3z))}{(4\cdot 2^{(\frac{x}{2})})})dxdydz\end{align*}$

Possible Answers:

2.22

-0.37

$-9.26\cdot10^{-2}$

$9.26\cdot10^{-2}$

Correct answer:

-0.37

Explanation:

$\begin{align*}&\int_{7}^{8.5}\int_{-4.5}^{-2}\int_{3}^{5.5}(\frac{(53\cdot 2^ycos(3z))}{(4\cdot 2^{(\frac{x}{2})})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{8.5}\int_{-4.5}^{-2}\int_{3}^{5.5}(\frac{(53\cdot 2^ycos(3z))}{(4\cdot 2^{(\frac{x}{2})})})dxdydz=\int_{7}^{8.5}\int_{-4.5}^{-2}(-\frac{(106\cdot 2^{(y - 2)}cos(3z))}{(2^{(\frac{x}{2})}ln(2))})dydz|_{3}^{5.5}\\&\int_{7}^{8.5}\int_{-4.5}^{-2}(\frac{(53\cdot 2^ycos(3z)\cdot (2\cdot 2^{(\frac{1}{2})} - 2^{(\frac{1}{4})}))}{(16ln(2))})dydz=\int_{7}^{8.5}(\frac{(53\cdot 2^y\cdot (2\cdot 2^{(\frac{1}{2})}cos(3z) - 2^{(\frac{1}{4})}cos(3z)))}{(16ln(2)^{2})})dz|_{-4.5}^{-2}\\&\int_{7}^{8.5}(\frac{(53cos(3z)\cdot (16\cdot 2^{(\frac{1}{2})} - 8\cdot 2^{(\frac{1}{4})} + 2^{(\frac{3}{4})} - 4))}{(512ln(2)^{2})})dz=\frac{(53sin(3z)\cdot (16\cdot 2^{(\frac{1}{2})} - 8\cdot 2^{(\frac{1}{4})} + 2^{(\frac{3}{4})} - 4))}{(1536ln(2)^{2})}|_{7}^{8.5}=-0.37\end{align*}$

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Example Question #452 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{10}\int_{8}^{9.5}\int_{-8}^{-4.5}(\frac{(5cos(3z)sin(x + 1))}{(37y)})dxdydz\end{align*}$

Possible Answers:

$6.20\cdot10^{-3}$

$-3.10\cdot10^{-3}$

$5.17\cdot10^{-4}$

$-1.86\cdot10^{-2}$

Correct answer:

$-3.10\cdot10^{-3}$

Explanation:

$\begin{align*}&\int_{6}^{10}\int_{8}^{9.5}\int_{-8}^{-4.5}(\frac{(5cos(3z)sin(x + 1))}{(37y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{10}\int_{8}^{9.5}\int_{-8}^{-4.5}(\frac{(5cos(3z)sin(x + 1))}{(37y)})dxdydz=\int_{6}^{10}\int_{8}^{9.5}(-\frac{(5cos(x + 1)cos(3z))}{(37y)})dydz|_{-8}^{-4.5}\\&\int_{6}^{10}\int_{8}^{9.5}(\frac{(5cos(3z)\cdot (cos(7) - cos(\frac{7}{2})))}{(37y)})dydz=\int_{6}^{10}(\frac{(5cos(3z)ln(y)\cdot (cos(7) - cos(\frac{7}{2})))}{37})dz|_{8}^{9.5}\\&\int_{6}^{10}(\frac{(5cos(3z)ln(\frac{19}{16})\cdot (cos(7) - cos(\frac{7}{2})))}{37})dz=\frac{(5sin(3z)ln(\frac{19}{16})\cdot (cos(7) - cos(\frac{7}{2})))}{111}|_{6}^{10}=-3.10\cdot10^{-3}\end{align*}$

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Example Question #454 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-3}\int_{10}^{14.5}\int_{6}^{8.5}(\frac{(3sin(y + 2))}{(496xz)})dxdydz\end{align*}$

Possible Answers:

$-2.26\cdot10^{-3}$

$2.26\cdot10^{-3}$

$7.53\cdot10^{-4}$

$-1.13\cdot10^{-2}$

Correct answer:

$-2.26\cdot10^{-3}$

Explanation:

$\begin{align*}&\int_{-6}^{-3}\int_{10}^{14.5}\int_{6}^{8.5}(\frac{(3sin(y + 2))}{(496xz)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-3}\int_{10}^{14.5}\int_{6}^{8.5}(\frac{(3sin(y + 2))}{(496xz)})dxdydz=\int_{-6}^{-3}\int_{10}^{14.5}(\frac{(3sin(y + 2)ln(x))}{(496z)})dydz|_{6}^{8.5}\\&\int_{-6}^{-3}\int_{10}^{14.5}(\frac{(3sin(y + 2)ln(\frac{17}{12}))}{(496z)})dydz=\int_{-6}^{-3}(-\frac{(3cos(y + 2)ln(\frac{17}{12}))}{(496z)})dz|_{10}^{14.5}\\&\int_{-6}^{-3}(\frac{(3ln(\frac{17}{12})\cdot (cos(12) - cos(\frac{33}{2})))}{(496z)})dz=ln(z)\cdot (\frac{(3cos(12)ln(\frac{17}{12}))}{496}-\frac{ (3cos(\frac{33}{2})ln(\frac{17}{12}))}{496})|_{-6}^{-3}=-2.26\cdot10^{-3}\end{align*}$

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Example Question #451 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-2}\int_{-3.5}^{1}\int_{-9}^{-5.5}(\frac{(29\cdot 3^{(\frac{y}{2})})}{(5x^{2}z^{2})})dxdydz\end{align*}$

Possible Answers:

$-7.05\cdot10^{-2}$

0.85

-0.85

0.42

Correct answer:

0.42

Explanation:

$\begin{align*}&\int_{-7}^{-2}\int_{-3.5}^{1}\int_{-9}^{-5.5}(\frac{(29\cdot 3^{(\frac{y}{2})})}{(5x^{2}z^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-2}\int_{-3.5}^{1}\int_{-9}^{-5.5}(\frac{(29\cdot 3^{(\frac{y}{2})})}{(5x^{2}z^{2})})dxdydz=\int_{-7}^{-2}\int_{-3.5}^{1}(-\frac{(29\cdot 3^{(\frac{y}{2})})}{(5xz^{2})})dydz|_{-9}^{-5.5}\\&\int_{-7}^{-2}\int_{-3.5}^{1}(\frac{(203\cdot 3^{(\frac{y}{2})})}{(495z^{2})})dydz=\int_{-7}^{-2}(\frac{(406\cdot 3^{(\frac{y}{2})})}{(495z^{2}ln(3))})dz|_{-3.5}^{1}\\&\int_{-7}^{-2}(\frac{(203\cdot (2\cdot 3^{(\frac{1}{2})} -\frac{ (2\cdot 3^{(\frac{1}{4})})}{9}))}{(495z^{2}ln(3))})dz=-\frac{(3654\cdot 3^{(\frac{1}{2})} - 406\cdot 3^{(\frac{1}{4})})}{(4455zln(3))}|_{-7}^{-2}=0.42\end{align*}$

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Example Question #456 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-4}\int_{3}^{7}\int_{4}^{9}(\frac{(9cos(4z))}{(53\cdot 3^{(\frac{x}{2})}y)})dxdydz\end{align*}$

Possible Answers:

$-1.05\cdot10^{-3}$

$7.01\cdot10^{-4}$

$4.21\cdot10^{-3}$

$-4.21\cdot10^{-3}$

Correct answer:

$-4.21\cdot10^{-3}$

Explanation:

$\begin{align*}&\int_{-6}^{-4}\int_{3}^{7}\int_{4}^{9}(\frac{(9cos(4z))}{(53\cdot 3^{(\frac{x}{2})}y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-4}\int_{3}^{7}\int_{4}^{9}(\frac{(9cos(4z))}{(53\cdot 3^{(\frac{x}{2})}y)})dxdydz=\int_{-6}^{-4}\int_{3}^{7}(-\frac{(18cos(4z))}{(53\cdot 3^{(\frac{x}{2})}yln(3))})dydz|_{4}^{9}\\&\int_{-6}^{-4}\int_{3}^{7}(-\frac{(cos(4z)\cdot (2\cdot 3^{(\frac{1}{2})} - 54))}{(1431yln(3))})dydz=\int_{-6}^{-4}(\frac{(ln(y)\cdot (54cos(4z) - 2\cdot 3^{(\frac{1}{2})}cos(4z)))}{(1431ln(3))})dz|_{3}^{7}\\&\int_{-6}^{-4}(\frac{(54cos(4z)ln(\frac{7}{3}) - 2\cdot 3^{(\frac{1}{2})}cos(4z)ln(\frac{7}{3}))}{(1431ln(3))})dz=\frac{(27sin(4z)ln(\frac{7}{3}) - 3^{(\frac{1}{2})}sin(4z)ln(\frac{7}{3}))}{(2862ln(3))}|_{-6}^{-4}=-4.21\cdot10^{-3}\end{align*}$

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Example Question #451 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{8}^{11.5}\int_{10}^{14.5}\int_{6}^{9.5}(\frac{(5sin(4x)sin(z + 1))}{(6y)})dxdydz\end{align*}$

Possible Answers:

-0.31

$-2.61\cdot10^{-2}$

0.16

$7.84\cdot10^{-2}$

Correct answer:

$7.84\cdot10^{-2}$

Explanation:

$\begin{align*}&\int_{8}^{11.5}\int_{10}^{14.5}\int_{6}^{9.5}(\frac{(5sin(4x)sin(z + 1))}{(6y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{8}^{11.5}\int_{10}^{14.5}\int_{6}^{9.5}(\frac{(5sin(4x)sin(z + 1))}{(6y)})dxdydz=\int_{8}^{11.5}\int_{10}^{14.5}(-\frac{(5cos(4x)sin(z + 1))}{(24y)})dydz|_{6}^{9.5}\\&\int_{8}^{11.5}\int_{10}^{14.5}(\frac{(5sin(z + 1)\cdot (\frac{cos(24)}{4}-\frac{ cos(38)}{4}))}{(6y)})dydz=\int_{8}^{11.5}(ln(y)\cdot (\frac{(5sin(z + 1)cos(24))}{24}-\frac{ (5sin(z + 1)cos(38))}{24}))dz|_{10}^{14.5}\\&\int_{8}^{11.5}(\frac{(5sin(z + 1)ln(\frac{29}{20})\cdot (cos(24) - cos(38)))}{24})dz=-\frac{(5cos(z + 1)ln(\frac{29}{20})\cdot (cos(24) - cos(38)))}{24}|_{8}^{11.5}=7.84\cdot10^{-2}\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #451 : Triple Integrals

Example Question #452 : Multiple Integration

Example Question #453 : Multiple Integration

Example Question #451 : Multiple Integration

Example Question #321 : Triple Integration In Cartesian Coordinates

Example Question #452 : Triple Integrals

Example Question #454 : Multiple Integration

Example Question #451 : Multiple Integration

Example Question #456 : Multiple Integration

Example Question #451 : Multiple Integration

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