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Calculus 3 : Multiple Integration

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Example Questions

Example Question #671 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(y^{2}sin(3z)\cdot(x +\frac{ 7}{25}))dxdydz\end{align*}$

Possible Answers:

9.51

-132.49

50.59

-3.56

Correct answer:

-3.56

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(y^{2}sin(3z)\cdot(x +\frac{ 7}{25}))dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(y^{2}sin(3z)\cdot(x +\frac{ 7}{25}))dxdydz=\int_{-3}^{2}\int_{-1}^{0}(\frac{(y^{2}sin(3z)\cdot(x +\frac{ 7}{25})^{2})}{2})dydz|_{2}^{6}\\&\int_{-3}^{2}\int_{-1}^{0}(\frac{(428y^{2}sin(3z))}{25})dydz=\int_{-3}^{2}(\frac{(428y^{3}sin(3z))}{75})dz|_{-1}^{0}\\&\int_{-3}^{2}(\frac{(428sin(3z))}{75})dz=-\frac{(428cos(3z))}{225}dz|_{-3}^{2}=-3.5596\end{align*}$

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Example Question #61 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{5}\int_{-4}^{2}\int_{-3}^{-1}(z + \frac{(29e^{({x + 2)}})}{12 }+ y^{2})dxdydz\end{align*}$

Possible Answers:

369.24

266.08

20143

136.08

Correct answer:

136.08

Explanation:

$\begin{align*}&\text{In performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{5}\int_{-4}^{2}\int_{-3}^{-1}(z + \frac{(29e^{({x + 2)}})}{12 }+ y^{2})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{5}\int_{-4}^{2}\int_{-3}^{-1}(z + \frac{(29e^{({x + 2)}})}{12 }+ y^{2})dxdydz=\int_{4}^{5}\int_{-4}^{2}(\frac{(29e^{({x + 2)}})}{12 }+ x\cdot(z + y^{2)})dydz|_{-3}^{-1}\\&\int_{4}^{5}\int_{-4}^{2}(2z - \frac{(29e^{({-1)}})}{12 }+ \frac{(29e^{({1)}})}{12 }+ 2y^{2})dydz=\int_{4}^{5}(y\cdot(2z - \frac{(29e^{({-1)}})}{12 }+ \frac{(29e^{({1)}})}{12}) + \frac{(2y^{3})}{3})dz|_{-4}^{2}\\&\int_{4}^{5}(12z - \frac{(29e^{({-1)}})}{2 }+ \frac{(29e^{({1)}})}{2 }+ 48)dz=\frac{(z\cdot(12z - 29e^{({-1)}} + 29e^{({1)}} + 96))}{2}dz|_{4}^{5}=136.08\end{align*}$

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Example Question #203 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-4}^{-1}\int_{5}^{10}\int_{-2}^{1}(\frac{(\frac{e^{({4x)}}}{6 }+ y^{2})}{z^{3}})dxdydz\end{align*}$

Possible Answers:

1.05

125.69

0.14

-415.49

Correct answer:

-415.49

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-4}^{-1}\int_{5}^{10}\int_{-2}^{1}(\frac{(\frac{e^{({4x)}}}{6 }+ y^{2})}{z^{3}})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4}^{-1}\int_{5}^{10}\int_{-2}^{1}(\frac{(\frac{e^{({4x)}}}{6 }+ y^{2})}{z^{3}})dxdydz=\int_{-4}^{-1}\int_{5}^{10}(\frac{(e^{({4x)}} + 24xy^{2})}{(24z^{3})})dydz|_{-2}^{1}\\&\int_{-4}^{-1}\int_{5}^{10}(\frac{(e^{({4)}} - e^{({-8)}} + 72y^{2})}{(24z^{3})})dydz=\int_{-4}^{-1}(\frac{(ye^{({-8)}}\cdot(e^{({12)}} + 24y^{2}e^{({8)}} - 1))}{(24z^{3})})dz|_{5}^{10}\\&\int_{-4}^{-1}(\frac{(5e^{({4)}} - 5e^{({-8)}} + 21000)}{(24z^{3})})dz=-\frac{(\frac{(5e^{({4)}})}{48 }- \frac{(5e^{({-8)}})}{48 }+\frac{ 875}{2})}{z^{2}}dz|_{-4}^{-1}=-415.49\end{align*}$

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Example Question #201 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-2}^{3}\int_{2}^{6}\int_{-1}^{3}(2\cdot3^ycos(4x))dxdydz\end{align*}$

Possible Answers:

-114.99

21.98

-2119.1

-63.31

Correct answer:

-2119.1

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-2}^{3}\int_{2}^{6}\int_{-1}^{3}(2\cdot3^ycos(4x))dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-2}^{3}\int_{2}^{6}\int_{-1}^{3}(2\cdot3^ycos(4x))dxdydz=\int_{-2}^{3}\int_{2}^{6}(\frac{(3^ysin(4x))}{2})dydz|_{-1}^{3}\\&\int_{-2}^{3}\int_{2}^{6}(3^y\cdot(\frac{sin(4)}{2}+\frac{ sin(12)}{2}))dydz=\int_{-2}^{3}(\frac{(3^y\cdot(sin(4) + sin(12)))}{(2ln(3))})dz|_{2}^{6}\\&\int_{-2}^{3}(\frac{(360\cdot(sin(4) + sin(12)))}{ln(3)})dz=\frac{(360z\cdot(sin(4) + sin(12)))}{ln(3)}dz|_{-2}^{3}=-2119.1\end{align*}$

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Example Question #202 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{9}\int_{-8}^{4}\int_{-3}^{1}(sin(z + 2)\cdot(3y +\frac{ (2\cdot3^{(x + 2)})}{53}))dxdydz\end{align*}$

Possible Answers:

-264.75

-41.9

41.9

Correct answer:

-264.75

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{9}\int_{-8}^{4}\int_{-3}^{1}(sin(z + 2)\cdot(3y +\frac{ (2\cdot3^{(x + 2)})}{53}))dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{9}\int_{-8}^{4}\int_{-3}^{1}(sin(z + 2)\cdot(3y +\frac{ (2\cdot3^{(x + 2))}}{53}))dxdydz=\int_{4}^{9}\int_{-8}^{4}(\frac{(3sin(z + 2)\cdot(6\cdot3^x + 53xyln(3)))}{(53ln(3))})dydz|_{-3}^{1}\\&\int_{4}^{9}\int_{-8}^{4}(\frac{(4sin(z + 2)\cdot(477yln(3) + 40))}{(159ln(3))})dydz=\int_{4}^{9}(\frac{(2sin(z + 2)\cdot(477yln(3) + 40)^{2})}{(75843ln(3)^{2})})dz|_{-8}^{4}\\&\int_{4}^{9}(-\frac{(32sin(z + 2)\cdot(477ln(3) - 20))}{(53ln(3))})dz=\frac{(32cos(z + 2)\cdot(477ln(3) - 20))}{(53ln(3))}dz|_{4}^{9}=-264.75\end{align*}$

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Example Question #203 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-2}^{1}\int_{3}^{5}\int_{-2}^{-1}(\frac{(43e^{(-2y)})}{(10x^{2})})dxdydz\end{align*}$

Possible Answers:

12.9

40.6

0.01

117.1

Correct answer:

0.01

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-2}^{1}\int_{3}^{5}\int_{-2}^{-1}(\frac{(43e^{(-2y)})}{(10x^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-2}^{1}\int_{3}^{5}\int_{-2}^{-1}(\frac{(43e^{(-2y)})}{(10x^{2})})dxdydz=\int_{-2}^{1}\int_{3}^{5}(-\frac{(43e^{(-2y)})}{(10x)})dydz|_{-2}^{-1}\\&\int_{-2}^{1}\int_{3}^{5}(\frac{(43e^{(-2y)})}{20})dydz=\int_{-2}^{1}(-\frac{(43e^{(-2y)})}{40})dz|_{3}^{5}\\&\int_{-2}^{1}(\frac{(43e^{(-10)}\cdot(e^{}(4) - 1))}{40})dz=\frac{(43ze^{(-10)}\cdot(e^{}(4) - 1))}{40}dz|_{-2}^{1}=0.0078476\end{align*}$

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Example Question #204 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-10}^{5}\int_{-4}^{1}\int_{-3}^{2}(\frac{(49\cdot2^zcos(3y)sin(4x))}{4})dxdydz\end{align*}$

Possible Answers:

-0.95

-18.44

-1.9

-0.41

Correct answer:

-18.44

Explanation:

$\begin{align*}&\text{Order of integration does not entirely matter.}\\&\text{Considering:}\\&\int_{-10}^{5}\int_{-4}^{1}\int_{-3}^{2}(\frac{(49\cdot2^zcos(3y)sin(4x))}{4})dxdydz\\&\text{Take it step by step:}\\&\int_{-10}^{5}\int_{-4}^{1}\int_{-3}^{2}(\frac{(49\cdot2^zcos(3y)sin(4x))}{4})dxdydz=\int_{-10}^{5}\int_{-4}^{1}(-\frac{(49\cdot2^zcos(4x)cos(3y))}{16})dydz|_{-3}^{2}\\&\int_{-10}^{5}\int_{-4}^{1}(-\frac{(49\cdot2^zcos(3y)\cdot(\frac{cos(8)}{4}-\frac{ cos(12)}{4}))}{4})dydz=\int_{-10}^{5}(-\frac{(49\cdot2^zsin(3y)\cdot(\frac{cos(8)}{4}-\frac{ cos(12)}{4}))}{12})dz|_{-4}^{1}\\&\int_{-10}^{5}(-\frac{(49\cdot2^z\cdot(sin(3) + sin(12))\cdot(cos(8) - cos(12)))}{48})dz=-\frac{(49\cdot2^z\cdot(\frac{sin(4)}{2}-\frac{ sin(5)}{2}+\frac{ sin(9)}{2}+\frac{ sin(11)}{2}-\frac{ sin(15)}{2}+\frac{ sin(20)}{2}-\frac{ sin(24)}{2}))}{(48ln(2))}dz|_{-10}^{5}=-18.44\end{align*}$

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Example Question #201 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-2}^{1}\int_{2}^{6}\int_{4}^{7}(\frac{(11x^{2})}{(35yz^{2})})dxdydz\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-2}^{1}\int_{2}^{6}\int_{4}^{7}(\frac{(11x^{2})}{(35yz^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-2}^{1}\int_{2}^{6}\int_{4}^{7}(\frac{(11x^{2})}{(35yz^{2})})dxdydz=\int_{-2}^{1}\int_{2}^{6}(\frac{(11x^{3})}{(105yz^{2})})dydz|_{4}^{7}\\&\int_{-2}^{1}\int_{2}^{6}(\frac{29.229}{(yz^{2})})dydz=\int_{-2}^{1}(\frac{(29.229ln(y))}{z^{2}})dz|_{2}^{6}\\&\int_{-2}^{1}(\frac{32.111}{z^{2}})dz=-\frac{32.111}{z}dz|_{-2}^{1}=Inf\end{align*}$

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Example Question #206 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{2}^{5}\int_{-6}^{-1}\int_{9}^{14}(\frac{(sin(3y)e^{(4x)})}{(6z)})dxdydz\end{align*}$

Possible Answers:

$-8.9522\cdot10^{22}$

$-1.3021\cdot10^{6}$

$4.393\cdot10^{22}$

Correct answer:

$4.393\cdot10^{22}$

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{5}\int_{-6}^{-1}\int_{9}^{14}(\frac{(sin(3y)e^{(4x)})}{(6z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{5}\int_{-6}^{-1}\int_{9}^{14}(\frac{(sin(3y)e^{(4x)})}{(6z)})dxdydz=\int_{2}^{5}\int_{-6}^{-1}(\frac{(sin(3y)e^{(4x)})}{(24z)})dydz|_{9}^{14}\\&\int_{2}^{5}\int_{-6}^{-1}(\frac{(8.7152\cdot10^{22}sin(3y))}{z})dydz=\int_{2}^{5}(-\frac{(2.9051\cdot10^{22}cos(3y))}{z})dz|_{-6}^{-1}\\&\int_{2}^{5}(\frac{4.7943\cdot10^{22}}{z})dz=4.7943\cdot10^{22}ln(z)dz|_{2}^{5}=4.393\cdot10^{22}\end{align*}$

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Example Question #207 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-8}^{-4}\int_{7}^{8}\int_{-10}^{-8}(3z + 2cos(3x)cos(y + 1))dxdydz\end{align*}$

Possible Answers:

28.77

-719.37

-143.87

431.62

Correct answer:

-143.87

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-8}^{-4}\int_{7}^{8}\int_{-10}^{-8}(3z + 2cos(3x)cos(y + 1))dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-8}^{-4}\int_{7}^{8}\int_{-10}^{-8}(3z + 2cos(3x)cos(y + 1))dxdydz=\int_{-8}^{-4}\int_{7}^{8}(3xz +\frac{ (2cos(y + 1)sin(3x))}{3})dydz|_{-10}^{-8}\\&\int_{-8}^{-4}\int_{7}^{8}(6z - 0.054969cos(y + 1))dydz=\int_{-8}^{-4}(6yz - 0.10994cos(0.5y + 0.5)sin(0.5y + 0.5))dz|_{7}^{8}\\&\int_{-8}^{-4}(6z + 0.03173)dz=0.03173z + 3z^{2}dz|_{-8}^{-4}=-143.87\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #671 : Calculus 3

Example Question #61 : Triple Integration In Cartesian Coordinates

Example Question #203 : Multiple Integration

Example Question #201 : Triple Integrals

Example Question #202 : Triple Integrals

Example Question #203 : Triple Integrals

Example Question #204 : Triple Integrals

Example Question #201 : Multiple Integration

Example Question #206 : Triple Integrals

Example Question #207 : Triple Integrals

All Calculus 3 Resources

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