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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

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:

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

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

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

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

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

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 #185 : Multiple Integration

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

Possible Answers:

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

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

40cos{(1/2)} + 40

40cos{(1/2)} - 50

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

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 #661 : Calculus 3

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

Possible Answers:

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

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

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

15cos{(5)} - 3885/4

15cos{(5)} - 15285/4

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 #181 : Multiple Integration

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

Possible Answers:

19539646797

1023

409064436

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 #188 : Multiple Integration

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

Possible Answers:

48700

10900

107500

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 #189 : Multiple Integration

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

Possible Answers:

18.52

-9.07

-117.87

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 #190 : Multiple Integration

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

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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Example Question #661 : Calculus 3

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

Possible Answers:

120

396

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 #191 : Multiple Integration

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

Possible Answers:

29.68

-182.63

17.68

-196.63

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 #192 : Multiple Integration

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

Possible Answers:

628.79

892.79

808.34

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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Calculus 3 : Triple Integration in Cartesian Coordinates

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #51 : Triple Integration In Cartesian Coordinates

Example Question #185 : Multiple Integration

Example Question #661 : Calculus 3

Example Question #181 : Multiple Integration

Example Question #188 : Multiple Integration

Example Question #189 : Multiple Integration

Example Question #190 : Multiple Integration

Example Question #661 : Calculus 3

Example Question #191 : Multiple Integration

Example Question #192 : Multiple Integration

All Calculus 3 Resources

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