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Calculus 3 : Multiple Integration

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Example Questions

Example Question #54 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-2}^{1}\int_{4}^{8}\int_{1}^{2}(6x - 2y)dxdydz\end{align*}$

Possible Answers:

396

120

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-2}^{1}\int_{4}^{8}\int_{1}^{2}(6x - 2y)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-2}^{1}\int_{4}^{8}\int_{1}^{2}(6x - 2y)dxdydz=\int_{-2}^{1}\int_{4}^{8}(x{(3x - 2y)})dydz|_{1}^{2}\\&\int_{-2}^{1}\int_{4}^{8}(9 - 2y)dydz=\int_{-2}^{1}(-y{(y - 9)})dz|_{4}^{8}\\&\int_{-2}^{1}(-12)dz=-12zdz|_{-2}^{1}=-36\end{align*}$

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Example Question #55 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

-182.63

17.68

-196.63

29.68

Correct answer:

-196.63

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{5}\int_{-3}^{-1}\int_{1}^{2}(x + 3y - e^{(z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{5}\int_{-3}^{-1}\int_{1}^{2}(x + 3y - e^{(z)})dxdydz=\int_{4}^{5}\int_{-3}^{-1}(x{(3y - e^{(z)})} + x^2/2)dydz|_{1}^{2}\\&\int_{4}^{5}\int_{-3}^{-1}(3y - e^{(z)} + 3/2)dydz=\int_{4}^{5}({(3y^2)}/2 - y{(e^{(z)} - 3/2)})dz|_{-3}^{-1}\\&\int_{4}^{5}(- 2e^{(z)} - 9)dz=- 9z - 2e^{(z)}dz|_{4}^{5}=-196.63\end{align*}$

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Example Question #56 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

808.34

628.79

892.79

700.34

Correct answer:

700.34

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{5}\int_{2}^{6}\int_{-5}^{-4}(e^{(z - 2)} + 3y^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{5}\int_{2}^{6}\int_{-5}^{-4}(e^{(z - 2)} + 3y^2)dxdydz=\int_{2}^{5}\int_{2}^{6}(x{(e^{(z - 2)} + 3y^2)})dydz|_{-5}^{-4}\\&\int_{2}^{5}\int_{2}^{6}(e^{(z - 2)} + 3y^2)dydz=\int_{2}^{5}(y^3 + ye^{(-2)}e^{(z)})dz|_{2}^{6}\\&\int_{2}^{5}(4e^{(-2)}e^{(z)} + 208)dz=208z + 4e^{(-2)}e^{(z)}dz|_{2}^{5}=700.34\end{align*}$

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Example Question #191 : Multiple Integration

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

Possible Answers:

112.11

34.01

32.11

226.01

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 #61 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

182.44

387.04

272.33

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 #61 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

6.43

495

32.14

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 #192 : Multiple Integration

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

Possible Answers:

3616

2096

1072

1568

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 #193 : Multiple Integration

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

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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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #54 : Triple Integration In Cartesian Coordinates

Example Question #55 : Triple Integration In Cartesian Coordinates

Example Question #56 : Triple Integration In Cartesian Coordinates

Example Question #191 : Multiple Integration

Example Question #61 : Triple Integration In Cartesian Coordinates

Example Question #61 : Triple Integration In Cartesian Coordinates

Example Question #61 : Triple Integration In Cartesian Coordinates

Example Question #191 : Multiple Integration

Example Question #192 : Multiple Integration

Example Question #193 : Multiple Integration

All Calculus 3 Resources

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