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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #192 : Multiple Integration

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

Possible Answers:

226.01

112.11

34.01

32.11

Correct answer:

32.11

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{5}^{9}\int_{3}^{4}\int_{-1}^{3}(2x + 4e^{(-z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{5}^{9}\int_{3}^{4}\int_{-1}^{3}(2x + 4e^{(-z)})dxdydz=\int_{5}^{9}\int_{3}^{4}(4xe^{(-z)} + x^2)dydz|_{-1}^{3}\\&\int_{5}^{9}\int_{3}^{4}(16e^{(-z)} + 8)dydz=\int_{5}^{9}(y{(16e^{(-z)} + 8)})dz|_{3}^{4}\\&\int_{5}^{9}(16e^{(-z)} + 8)dz=8z - 16e^{(-z)}dz|_{5}^{9}=32.106\end{align*}$

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Example Question #195 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{0}^{3}\int_{-1}^{2}\int_{0}^{3}(10y - 4e^{(-2z)})dxdydz\end{align*}$

Possible Answers:

182.44

272.33

387.04

117.04

Correct answer:

117.04

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{0}^{3}\int_{-1}^{2}\int_{0}^{3}(10y - 4e^{(-2z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{0}^{3}\int_{-1}^{2}\int_{0}^{3}(10y - 4e^{(-2z)})dxdydz=\int_{0}^{3}\int_{-1}^{2}(x{(10y - 4e^{(-2z)})})dydz|_{0}^{3}\\&\int_{0}^{3}\int_{-1}^{2}(30y - 12e^{(-2z)})dydz=\int_{0}^{3}(15y^2 - 12ye^{(-2z)})dz|_{-1}^{2}\\&\int_{0}^{3}(45 - 36e^{(-2z)})dz=45z + 18e^{(-2z)}dz|_{0}^{3}=117.04\end{align*}$

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Example Question #61 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{6}\int_{2}^{4}\int_{3}^{7}(8ye^{(-2z)})dxdydz\end{align*}$

Possible Answers:

4.32

3.88

0.4

0.24

Correct answer:

0.24

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{3}^{6}\int_{2}^{4}\int_{3}^{7}(8ye^{(-2z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{6}\int_{2}^{4}\int_{3}^{7}(8ye^{(-2z)})dxdydz=\int_{3}^{6}\int_{2}^{4}(8xye^{(-2z)})dydz|_{3}^{7}\\&\int_{3}^{6}\int_{2}^{4}(32ye^{(-2z)})dydz=\int_{3}^{6}(16y^2e^{(-2z)})dz|_{2}^{4}\\&\int_{3}^{6}(192e^{(-2z)})dz=-96e^{(-2z)}dz|_{3}^{6}=0.23737\end{align*}$

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Example Question #197 : Multiple Integration

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

Possible Answers:

495

32.14

6.43

Correct answer:

495

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{1}^{4}\int_{4}^{7}\int_{-2}^{3}({(8y)}/z^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{1}^{4}\int_{4}^{7}\int_{-2}^{3}({(8y)}/z^2)dxdydz=\int_{1}^{4}\int_{4}^{7}({(8xy)}/z^2)dydz|_{-2}^{3}\\&\int_{1}^{4}\int_{4}^{7}({(40y)}/z^2)dydz=\int_{1}^{4}({(20y^2)}/z^2)dz|_{4}^{7}\\&\int_{1}^{4}(660/z^2)dz=-660/zdz|_{1}^{4}=495\end{align*}$

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Example Question #191 : Multiple Integration

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

Possible Answers:

123.2

78.4

51.73

Correct answer:

123.2

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{5}^{9}\int_{3}^{4}\int_{-5}^{-3}(2z + {(6y)}/x^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{5}^{9}\int_{3}^{4}\int_{-5}^{-3}(2z + {(6y)}/x^2)dxdydz=\int_{5}^{9}\int_{3}^{4}(2xz - {(6y)}/x)dydz|_{-5}^{-3}\\&\int_{5}^{9}\int_{3}^{4}({(4y)}/5 + 4z)dydz=\int_{5}^{9}({(2y{(y + 10z)})}/5)dz|_{3}^{4}\\&\int_{5}^{9}(4z + 14/5)dz={(2z{(5z + 7)})}/5dz|_{5}^{9}=123.2\end{align*}$

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Example Question #671 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{2}^{4}\int_{0}^{4}\int_{0}^{2}(z + 24xy^2)dxdydz\end{align*}$

Possible Answers:

2096

1072

1568

3616

Correct answer:

2096

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{4}\int_{0}^{4}\int_{0}^{2}(z + 24xy^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{4}\int_{0}^{4}\int_{0}^{2}(z + 24xy^2)dxdydz=\int_{2}^{4}\int_{0}^{4}(x{(z + 12xy^2)})dydz|_{0}^{2}\\&\int_{2}^{4}\int_{0}^{4}(2z + 48y^2)dydz=\int_{2}^{4}(2yz + 16y^3)dz|_{0}^{4}\\&\int_{2}^{4}(8z + 1024)dz=4z{(z + 256)}dz|_{2}^{4}=2096\end{align*}$

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Example Question #63 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{5}^{9}\int_{-5}^{0}\int_{-1}^{4}(12xy^2 - 3z^2)dxdydz\end{align*}$

Possible Answers:

33900

88100

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{5}^{9}\int_{-5}^{0}\int_{-1}^{4}(12xy^2 - 3z^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{5}^{9}\int_{-5}^{0}\int_{-1}^{4}(12xy^2 - 3z^2)dxdydz=\int_{5}^{9}\int_{-5}^{0}(3x{(2xy^2 - z^2)})dydz|_{-1}^{4}\\&\int_{5}^{9}\int_{-5}^{0}(90y^2 - 15z^2)dydz=\int_{5}^{9}(30y^3 - 15yz^2)dz|_{-5}^{0}\\&\int_{5}^{9}(3750 - 75z^2)dz=-25z{(z^2 - 150)}dz|_{5}^{9}=-100\end{align*}$

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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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Calculus 3 : Triple Integration in Cartesian Coordinates

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #192 : Multiple Integration

Example Question #195 : Multiple Integration

Example Question #61 : Triple Integration In Cartesian Coordinates

Example Question #197 : Multiple Integration

Example Question #191 : Multiple Integration

Example Question #671 : Calculus 3

Example Question #63 : Triple Integration In Cartesian Coordinates

Example Question #671 : Calculus 3

Example Question #61 : Triple Integration In Cartesian Coordinates

Example Question #203 : Multiple Integration

All Calculus 3 Resources

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