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Calculus 3 : Multiple Integration

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Example Questions

Example Question #101 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{1}^{6}\int_{6}^{8}\int_{10}^{12}(\frac{(2\cdot2^z\cdot4^ycos(x + 2))}{25})dxdydz\end{align*}$

Possible Answers:

$-2.422\cdot10^{6}$

$1.937\cdot10^{6}$

484300

Correct answer:

484300

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{1}^{6}\int_{6}^{8}\int_{10}^{12}(\frac{(2\cdot2^z\cdot4^ycos(x + 2))}{25})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{1}^{6}\int_{6}^{8}\int_{10}^{12}(\frac{(2\cdot2^z\cdot4^ycos(x + 2))}{25})dxdydz=\int_{1}^{6}\int_{6}^{8}(\frac{(2\cdot2^{(2y + z)}sin(x + 2))}{25})dydz|_{10}^{12}\\&\int_{1}^{6}\int_{6}^{8}(-2^{(2y + z)}\cdot(\frac{(2sin(12))}{25}-\frac{ (2sin(14))}{25}))dydz=\int_{1}^{6}(-\frac{(2^{(2y + z)}\cdot(sin(12) - sin(14)))}{(25ln(2))})dz|_{6}^{8}\\&\int_{1}^{6}(-\frac{(12288\cdot2^z\cdot(sin(12) - sin(14)))}{(5ln(2))})dz=-\frac{(12288\cdot2^z\cdot(sin(12) - sin(14)))}{(5ln(2)^{2})}|_{1}^{6}=484300\end{align*}$

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Example Question #102 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-4}\int_{6}^{8}\int_{9}^{14}(\frac{(3\cdot4^{(\frac{z}{2})}cos(x + 2))}{(2y)})dxdydz\end{align*}$

Possible Answers:

0.02

0.15

-0.01

-0.15

Correct answer:

0.02

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-7}^{-4}\int_{6}^{8}\int_{9}^{14}(\frac{(3\cdot4^{(\frac{z}{2})}cos(x + 2))}{(2y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-4}\int_{6}^{8}\int_{9}^{14}(\frac{(3\cdot4^{(\frac{z}{2})}cos(x + 2))}{(2y)})dxdydz=\int_{-7}^{-4}\int_{6}^{8}(\frac{(3\cdot2^zsin(x + 2))}{(2y)})dydz|_{9}^{14}\\&\int_{-7}^{-4}\int_{6}^{8}(-\frac{(2^z\cdot(\frac{(3sin(11))}{2}-\frac{ (3sin(16))}{2}))}{y})dydz=\int_{-7}^{-4}(-\frac{(3\cdot2^zln(y)\cdot(sin(11) - sin(16)))}{2})dz|_{6}^{8}\\&\int_{-7}^{-4}(-\frac{(3\cdot2^zln(\frac{4}{3})\cdot(sin(11) - sin(16)))}{2})dz=-\frac{(3\cdot2^z\cdot(ln(\frac{4}{3})sin(11) - ln(\frac{4}{3})sin(16)))}{(2ln(2))}|_{-7}^{-4}=0.02\end{align*}$

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Example Question #110 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{8}\int_{-8}^{-3}\int_{6}^{9}(\frac{(17e^{(\frac{x}{3})})}{(6\cdot2^y\cdot4^{(\frac{z}{2})})})dxdydz\end{align*}$

Possible Answers:

3264

9792

-652.8

Correct answer:

3264

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{8}\int_{-8}^{-3}\int_{6}^{9}(\frac{(17e^{(\frac{x}{3})})}{(6\cdot2^y\cdot4^{(\frac{z}{2})})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{8}\int_{-8}^{-3}\int_{6}^{9}(\frac{(17e^{(\frac{x}{3})})}{(6\cdot2^y\cdot4^{(\frac{z}{2})})})dxdydz=\int_{4}^{8}\int_{-8}^{-3}(\frac{(17\cdot(\frac{1}{2})^{(y + z)}e^{(\frac{x}{3})})}{2})dydz|_{6}^{9}\\&\int_{4}^{8}\int_{-8}^{-3}(\frac{(17\cdot(\frac{1}{2})^{(y + z)}e^{(2)}\cdot(e^{(1)} - 1))}{2})dydz=\int_{4}^{8}(\frac{(\frac{17}{2^{(y}+ z)}\cdot(e^{(2)} - e^{(3)}))}{(2ln(2))})dz|_{-8}^{-3}\\&\int_{4}^{8}(\frac{(2108e^{(2)}\cdot(e^{(1)} - 1))}{(2^zln(2))})dz=\frac{(2108\cdot(e^{(2)} - e^{(3)}))}{(2^zln(2)^{2})}|_{4}^{8}=3264\end{align*}$

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Example Question #721 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{8}\int_{7}^{9}\int_{8}^{13}(\frac{(33\cdot3^{(\frac{z}{2})}sin(x + 2)e^{(\frac{y}{2})})}{4})dxdydz\end{align*}$

Possible Answers:

2442

39088

Correct answer:

Explanation:

$\begin{align*}\\&\int_{4}^{8}\int_{7}^{9}\int_{8}^{13}(\frac{(33\cdot3^{(\frac{z}{2})}sin(x + 2)e^{(\frac{y}{2})})}{4})dxdydz=\int_{4}^{8}\int_{7}^{9}(-\frac{(33\cdot3^{(\frac{z}{2})}cos(x + 2)e^{(\frac{y}{2})})}{4})dydz|_{8}^{13}\\&\int_{4}^{8}\int_{7}^{9}(\frac{(33\cdot3^{(\frac{z}{2})}e^{(\frac{y}{2})}\cdot(cos(10) - cos(15)))}{4})dydz=\int_{4}^{8}(\frac{(33\cdot3^{(\frac{z}{2})}e^{(\frac{y}{2})}\cdot(cos(10) - cos(15)))}{2})dz|_{7}^{9}\\&\int_{4}^{8}(\frac{(33\cdot3^{(\frac{z}{2})}e^{(\frac{7}{2})}\cdot(cos(10) - cos(15))\cdot(e^{(1)} - 1))}{2})dz=-\frac{(33\cdot3^{(\frac{z}{2})}\cdot(cos(10)e^{(\frac{7}{2})} - cos(10)e^{(\frac{9}{2})} - cos(15)e^{(\frac{7}{2})} + cos(15)e^{(\frac{9}{2})}))}{ln(3)}|_{4}^{8}=-9769\end{align*}$

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Example Question #241 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-2}\int_{8}^{13}\int_{-8}^{-6}(\frac{(2cos(x + 1)cos(y + 2))}{(33z)})dxdydz\end{align*}$

Possible Answers:

-0.29

-0.15

0.07

0.29

Correct answer:

-0.15

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-7}^{-2}\int_{8}^{13}\int_{-8}^{-6}(\frac{(2cos(x + 1)cos(y + 2))}{(33z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-2}\int_{8}^{13}\int_{-8}^{-6}(\frac{(2cos(x + 1)cos(y + 2))}{(33z)})dxdydz=\int_{-7}^{-2}\int_{8}^{13}(\frac{(2cos(y + 2)sin(x + 1))}{(33z)})dydz|_{-8}^{-6}\\&\int_{-7}^{-2}\int_{8}^{13}(-\frac{(2cos(y + 2)\cdot(sin(5) - sin(7)))}{(33z)})dydz=\int_{-7}^{-2}(-\frac{(2sin(y + 2)\cdot(sin(5) - sin(7)))}{(33z)})dz|_{8}^{13}\\&\int_{-7}^{-2}(\frac{(2\cdot(sin(5) - sin(7))\cdot(sin(10) - sin(15)))}{(33z)})dz=ln(z)\cdot(\frac{(2sin(5)sin(10))}{33}-\frac{ (2sin(7)sin(10))}{33}-\frac{ (2sin(5)sin(15))}{33}+\frac{ (2sin(7)sin(15))}{33})|_{-7}^{-2}=-0.15\end{align*}$

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Example Question #242 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-8}^{-7}\int_{8}^{10}\int_{7}^{12}(\frac{(25sin(3z))}{x})dxdydz\end{align*}$

Possible Answers:

-1.46

17.46

8.73

-34.92

Correct answer:

8.73

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-8}^{-7}\int_{8}^{10}\int_{7}^{12}(\frac{(25sin(3z))}{x})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-8}^{-7}\int_{8}^{10}\int_{7}^{12}(\frac{(25sin(3z))}{x})dxdydz=\int_{-8}^{-7}\int_{8}^{10}(25sin(3z)ln(x))dydz|_{7}^{12}\\&\int_{-8}^{-7}\int_{8}^{10}(25sin(3z)ln(\frac{12}{7}))dydz=\int_{-8}^{-7}(25ysin(3z)ln(\frac{12}{7}))dz|_{8}^{10}\\&\int_{-8}^{-7}(50sin(3z)ln(\frac{12}{7}))dz=-\frac{(50cos(3z)ln(\frac{12}{7}))}{3}|_{-8}^{-7}=8.73\end{align*}$

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Example Question #243 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-6}\int_{7}^{10}\int_{6}^{9}(\frac{(5ysin(z + 1))}{(3x)})dxdydz\end{align*}$

Possible Answers:

1.48

-7.39

-36.98

36.98

Correct answer:

-7.39

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-9}^{-6}\int_{7}^{10}\int_{6}^{9}(\frac{(5ysin(z + 1))}{(3x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-6}\int_{7}^{10}\int_{6}^{9}(\frac{(5ysin(z + 1))}{(3x)})dxdydz=\int_{-9}^{-6}\int_{7}^{10}(\frac{(5ysin(z + 1)ln(x))}{3})dydz|_{6}^{9}\\&\int_{-9}^{-6}\int_{7}^{10}(\frac{(5ysin(z + 1)ln(\frac{3}{2}))}{3})dydz=\int_{-9}^{-6}(\frac{(5y^{2}sin(z + 1)ln(\frac{3}{2}))}{6})dz|_{7}^{10}\\&\int_{-9}^{-6}(\frac{(85sin(z + 1)ln(\frac{3}{2}))}{2})dz=-\frac{(85cos(z + 1)ln(\frac{3}{2}))}{2}|_{-9}^{-6}=-7.39\end{align*}$

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Example Question #244 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-6}\int_{6}^{9}\int_{10}^{15}(\frac{(11e^{(-\frac{x}{2})})}{(8z)})dxdydz\end{align*}$

Possible Answers:

-0.1

-0.02

0.04

Correct answer:

-0.02

Explanation:

$\begin{align*}&\int_{-9}^{-6}\int_{6}^{9}\int_{10}^{15}(\frac{(11e^{(-\frac{x}{2})})}{(8z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-6}\int_{6}^{9}\int_{10}^{15}(\frac{(11e^{(-\frac{x}{2})})}{(8z)})dxdydz=\int_{-9}^{-6}\int_{6}^{9}(-\frac{(11e^{(-\frac{x}{2})})}{(4z)})dydz|_{10}^{15}\\&\int_{-9}^{-6}\int_{6}^{9}(\frac{(11e^{(-\frac{15}{2})}\cdot(e^{(\frac{5}{2})} - 1))}{(4z)})dydz=\int_{-9}^{-6}(\frac{(11ye^{(-\frac{15}{2})}\cdot(e^{(\frac{5}{2})} - 1))}{(4z)})dz|_{6}^{9}\\&\int_{-9}^{-6}(\frac{(33e^{(-\frac{15}{2})}\cdot(e^{(\frac{5}{2})} - 1))}{(4z)})dz=\frac{(33e^{(-\frac{15}{2})}ln(z)\cdot(e^{(\frac{5}{2})} - 1))}{4}|_{-9}^{-6}=-0.02\end{align*}$

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Example Question #245 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{1}^{3}\int_{-8}^{4}\int_{-8}^{-6}(\frac{(5sin(y + 1))}{(33z)})dxdydz\end{align*}$

Possible Answers:

-0.08

0.16

0.47

-0.31

Correct answer:

0.16

Explanation:

$\begin{align*}&\int_{1}^{3}\int_{-8}^{4}\int_{-8}^{-6}(\frac{(5sin(y + 1))}{(33z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{1}^{3}\int_{-8}^{4}\int_{-8}^{-6}(\frac{(5sin(y + 1))}{(33z)})dxdydz=\int_{1}^{3}\int_{-8}^{4}(\frac{(5xsin(y + 1))}{(33z)})dydz|_{-8}^{-6}\\&\int_{1}^{3}\int_{-8}^{4}(\frac{(10sin(y + 1))}{(33z)})dydz=\int_{1}^{3}(-\frac{(10cos(y + 1))}{(33z)})dz|_{-8}^{4}\\&\int_{1}^{3}(-\frac{(10\cdot(cos(5) - cos(7)))}{(33z)})dz=-ln(z)\cdot(\frac{(10cos(5))}{33}-\frac{ (10cos(7))}{33})|_{1}^{3}=0.16\end{align*}$

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Example Question #241 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{12}\int_{6}^{9}\int_{-8}^{-4}(\frac{(5cos(x + 2)e^{(\frac{y}{2})}e^{(z)})}{9})dxdydz\end{align*}$

Possible Answers:

$-1.428\cdot10^{7}$

$-2.857\cdot10^{7}$

$5.714\cdot10^{7}$

$2.857\cdot10^{6}$

Correct answer:

$-1.428\cdot10^{7}$

Explanation:

$\begin{align*}&\int_{9}^{12}\int_{6}^{9}\int_{-8}^{-4}(\frac{(5cos(x + 2)e^{(\frac{y}{2})}e^{(z)})}{9})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{12}\int_{6}^{9}\int_{-8}^{-4}(\frac{(5cos(x + 2)e^{(\frac{y}{2})}e^{(z)})}{9})dxdydz=\int_{9}^{12}\int_{6}^{9}(\frac{(5sin(x + 2)e^{(\frac{y}{2}+ z)})}{9})dydz|_{-8}^{-4}\\&\int_{9}^{12}\int_{6}^{9}(-e^{(\frac{y}{2}+ z)}\cdot(\frac{(5sin(2))}{9}-\frac{ (5sin(6))}{9}))dydz=\int_{9}^{12}(-\frac{(10e^{(\frac{y}{2}+ z)}\cdot(sin(2) - sin(6)))}{9})dz|_{6}^{9}\\&\int_{9}^{12}(-\frac{(10e^{(3)}e^{(z)}\cdot(sin(2) - sin(6))\cdot(e^{(\frac{3}{2})} - 1))}{9})dz=-\frac{(10e^{(z + 3)}\cdot(sin(2) - sin(6))\cdot(e^{(\frac{3}{2})} - 1))}{9}|_{9}^{12}=-1.428\cdot10^{7}\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #101 : Triple Integration In Cartesian Coordinates

Example Question #102 : Triple Integration In Cartesian Coordinates

Example Question #110 : Triple Integration In Cartesian Coordinates

Example Question #721 : Calculus 3

Example Question #241 : Multiple Integration

Example Question #242 : Multiple Integration

Example Question #243 : Multiple Integration

Example Question #244 : Multiple Integration

Example Question #245 : Multiple Integration

Example Question #241 : Triple Integrals

All Calculus 3 Resources

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