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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #41 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{2}^{3}\int_{-2}^{3}\int_{2}^{6}(xy^2)dxdydz\end{align*}$

Possible Answers:

2600/3

560/3

350/3

520/3

1520/3

Correct answer:

560/3

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{3}\int_{-2}^{3}\int_{2}^{6}(xy^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{3}\int_{-2}^{3}\int_{2}^{6}(xy^2)dxdydz=\int_{2}^{3}\int_{-2}^{3}({(x^2y^2)}/2)dydz|_{2}^{6}\\&\int_{2}^{3}\int_{-2}^{3}(16y^2)dydz=\int_{2}^{3}({(16y^3)}/3)dz|_{-2}^{3}\\&\int_{2}^{3}(560/3)dz={(560z)}/3dz|_{2}^{3}=560/3\end{align*}$

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Example Question #46 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{1}^{3}\int_{5}^{7}\int_{-4}^{-1}(z - 3xy^2)dxdydz\end{align*}$

Possible Answers:

462

3294

Correct answer:

3294

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{1}^{3}\int_{5}^{7}\int_{-4}^{-1}(z - 3xy^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{1}^{3}\int_{5}^{7}\int_{-4}^{-1}(z - 3xy^2)dxdydz=\int_{1}^{3}\int_{5}^{7}(xz - {(3x^2y^2)}/2)dydz|_{-4}^{-1}\\&\int_{1}^{3}\int_{5}^{7}(3z + {(45y^2)}/2)dydz=\int_{1}^{3}(3yz + {(15y^3)}/2)dz|_{5}^{7}\\&\int_{1}^{3}(6z + 1635)dz=3z{(z + 545)}dz|_{1}^{3}=3294\end{align*}$

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Example Question #44 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{2}\int_{3}^{7}\int_{5}^{7}(y - e^{(z)})dxdydz\end{align*}$

Possible Answers:

$8e^{(-1)} - 8e^{(2)} + 144$

$6e^{(3)} - 6e^{(7)} + 12$

$8e^{(-1)} - 8e^{(2)} + 120$

$6e^{(3)} - 6e^{(7)} + 144$

$12e^{(5)} - 12e^{(7)} + 120$

Correct answer:

$8e^{(-1)} - 8e^{(2)} + 120$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{2}\int_{3}^{7}\int_{5}^{7}(y - e^{(z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{2}\int_{3}^{7}\int_{5}^{7}(y - e^{(z)})dxdydz=\int_{-1}^{2}\int_{3}^{7}(x{(y - e^{(z)})})dydz|_{5}^{7}\\&\int_{-1}^{2}\int_{3}^{7}(2y - 2e^{(z)})dydz=\int_{-1}^{2}(y^2 - 2ye^{(z)})dz|_{3}^{7}\\&\int_{-1}^{2}(40 - 8e^{(z)})dz=40z - 8e^{(z)}dz|_{-1}^{2}=8e^{(-1)} - 8e^{(2)} + 120\end{align*}$

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Example Question #51 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{5}^{6}\int_{0}^{4}\int_{-5}^{-4}(e^{(y/2)} - e^{(z)})dxdydz\end{align*}$

Possible Answers:

$8e^{(-5/2)}{(e^{(1/2)} - 1)} - e^{(4)} + 1$

$2e^{(2)} - 4e^{(-4)} + 4e^{(-5)} - 2$

$8e^{(5/2)}{(e^{(1/2)} - 1)} - e^{(4)} + 1$

$2e^{(2)} + 4e^{(5)} - 4e^{(6)} - 2$

$4e^{(5)} - 4e^{(6)} + 8e^{(-5/2)}{(e^{(1/2)} - 1)}$

Correct answer:

$2e^{(2)} + 4e^{(5)} - 4e^{(6)} - 2$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{5}^{6}\int_{0}^{4}\int_{-5}^{-4}(e^{(y/2)} - e^{(z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{5}^{6}\int_{0}^{4}\int_{-5}^{-4}(e^{(y/2)} - e^{(z)})dxdydz=\int_{5}^{6}\int_{0}^{4}(x{(e^{(y/2)} - e^{(z)})})dydz|_{-5}^{-4}\\&\int_{5}^{6}\int_{0}^{4}(e^{(y/2)} - e^{(z)})dydz=\int_{5}^{6}(2e^{(y/2)} - ye^{(z)})dz|_{0}^{4}\\&\int_{5}^{6}(2e^{(2)} - 4e^{(z)} - 2)dz=z{(2e^{(2)} - 2)} - 4e^{(z)}dz|_{5}^{6}=2e^{(2)} + 4e^{(5)} - 4e^{(6)} - 2\end{align*}$

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Example Question #52 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

40cos{(1/2)} + 40

10cos{(1)} - 10cos{(3)} + 10

8cos{(3/2)} - 8cos{(1)} + 80

10cos{(1)} - 10cos{(3)} - 10

40cos{(1/2)} - 50

Correct answer:

40cos{(1/2)} + 40

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(x + sin{(y/2)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(x + sin{(y/2)})dxdydz=\int_{-3}^{2}\int_{-1}^{0}({(x{(x + 2sin{(y/2)})})}/2)dydz|_{2}^{6}\\&\int_{-3}^{2}\int_{-1}^{0}(4sin{(y/2)} + 16)dydz=\int_{-3}^{2}(16y - 8cos{(y/2)})dz|_{-1}^{0}\\&\int_{-3}^{2}(8cos{(1/2)} + 8)dz=z{(8cos{(1/2)} + 8)}dz|_{-3}^{2}=40cos{(1/2)} + 40\end{align*}$

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Example Question #52 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

15cos{(5)} - 15285/4

25cos{(2)} - 25cos{(5)} + 9375/4

15cos{(1)} - 15cos{(4)} + 9375/4

25cos{(2)} - 25cos{(5)} - 3825/4

15cos{(5)} - 3885/4

Correct answer:

25cos{(2)} - 25cos{(5)} + 9375/4

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-5}^{-2}\int_{0}^{5}\int_{-4}^{1}(y^3 - sin{(z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-5}^{-2}\int_{0}^{5}\int_{-4}^{1}(y^3 - sin{(z)})dxdydz=\int_{-5}^{-2}\int_{0}^{5}(-x{(sin{(z)} - y^3)})dydz|_{-4}^{1}\\&\int_{-5}^{-2}\int_{0}^{5}(5y^3 - 5sin{(z)})dydz=\int_{-5}^{-2}({(5y^4)}/4 - 5ysin{(z)})dz|_{0}^{5}\\&\int_{-5}^{-2}(3125/4 - 25sin{(z)})dz={(3125z)}/4 + 25cos{(z)}dz|_{-5}^{-2}=25cos{(2)} - 25cos{(5)} + 9375/4\end{align*}$

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Example Question #54 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-2}^{1}\int_{5}^{7}\int_{9}^{10}(5z^9)dxdydz\end{align*}$

Possible Answers:

19539646797

409064436

1023

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-2}^{1}\int_{5}^{7}\int_{9}^{10}(5z^9)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-2}^{1}\int_{5}^{7}\int_{9}^{10}(5z^9)dxdydz=\int_{-2}^{1}\int_{5}^{7}(5xz^9)dydz|_{9}^{10}\\&\int_{-2}^{1}\int_{5}^{7}(5z^9)dydz=\int_{-2}^{1}(5yz^9)dz|_{5}^{7}\\&\int_{-2}^{1}(10z^9)dz=z^{10}dz|_{-2}^{1}=-1023\end{align*}$

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Example Question #51 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{1}^{2}\int_{-4}^{1}\int_{3}^{8}(12{(3x + 4y)}^2)dxdydz\end{align*}$

Possible Answers:

48700

107500

10900

220900

157900

Correct answer:

48700

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{1}^{2}\int_{-4}^{1}\int_{3}^{8}(12{(3x + 4y)}^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{1}^{2}\int_{-4}^{1}\int_{3}^{8}(12{(3x + 4y)}^2)dxdydz=\int_{1}^{2}\int_{-4}^{1}({(4{(3x + 4y)}^3)}/3)dydz\Big|_{3}^{8}\\&\int_{1}^{2}\int_{-4}^{1}(7920y + 960y^2 + 17460)dydz=\int_{1}^{2}(20y{(198y + 16y^2 + 873)})\Big|_{-4}^{1}\\&\int_{1}^{2}(48700)dz=48700zdz\Big|_{1}^{2}=48700\end{align*}$

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Example Question #52 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(x^3e^{(y - 4)})dxdydz\end{align*}$

Possible Answers:

-117.87

18.52

-9.07

43.02

Correct answer:

18.52

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(x^3e^{(y - 4)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(x^3e^{(y - 4)})dxdydz=\int_{-3}^{2}\int_{-1}^{0}(0.004579x^4e^{(y)})dydz|_{2}^{6}\\&\int_{-3}^{2}\int_{-1}^{0}(5.861e^{(y)})dydz=\int_{-3}^{2}(5.861e^{(y)})dz|_{-1}^{0}\\&\int_{-3}^{2}(3.7049)dz=3.705zdz|_{-3}^{2}=18.524\end{align*}$

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Example Question #53 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{4}\int_{-3}^{2}\int_{-5}^{-3}(4x + 2y)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{4}\int_{-3}^{2}\int_{-5}^{-3}(4x + 2y)dxdydz=\int_{-1}^{4}\int_{-3}^{2}(2x{(x + y)})dydz|_{-5}^{-3}\\&\int_{-1}^{4}\int_{-3}^{2}(4y - 32)dydz=\int_{-1}^{4}(2y{(y - 16)})dz|_{-3}^{2}\\&\int_{-1}^{4}(-170)dz=-170zdz|_{-1}^{4}=-850\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #41 : Triple Integration In Cartesian Coordinates

Example Question #46 : Triple Integration In Cartesian Coordinates

Example Question #44 : Triple Integration In Cartesian Coordinates

Example Question #51 : Triple Integration In Cartesian Coordinates

Example Question #52 : Triple Integration In Cartesian Coordinates

Example Question #52 : Triple Integration In Cartesian Coordinates

Example Question #54 : Triple Integration In Cartesian Coordinates

Example Question #51 : Triple Integration In Cartesian Coordinates

Example Question #52 : Triple Integration In Cartesian Coordinates

Example Question #53 : Triple Integration In Cartesian Coordinates

All Calculus 3 Resources

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