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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #41 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

1580

680

1220

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-3}^{-2}\int_{-3}^{1}\int_{1}^{6}(4y - 4x^3)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{-2}\int_{-3}^{1}\int_{1}^{6}(4y - 4x^3)dxdydz=\int_{-3}^{-2}\int_{-3}^{1}(4xy - x^4)dydz|_{1}^{6}\\&\int_{-3}^{-2}\int_{-3}^{1}(20y - 1295)dydz=\int_{-3}^{-2}(5y{(2y - 259)})dz|_{-3}^{1}\\&\int_{-3}^{-2}(-5260)dz=-5260zdz|_{-3}^{-2}=-5260\end{align*}$

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Example Question #42 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{5}\int_{0}^{4}\int_{1}^{4}(80x - 120z^2)dxdydz\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{3}^{5}\int_{0}^{4}\int_{1}^{4}(80x - 120z^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{5}\int_{0}^{4}\int_{1}^{4}(80x - 120z^2)dxdydz=\int_{3}^{5}\int_{0}^{4}(40x{(x - 3z^2)})dydz|_{1}^{4}\\&\int_{3}^{5}\int_{0}^{4}(600 - 360z^2)dydz=\int_{3}^{5}(-y{(360z^2 - 600)})dz|_{0}^{4}\\&\int_{3}^{5}(2400 - 1440z^2)dz=-480z{(z^2 - 5)}dz|_{3}^{5}=-42240\end{align*}$

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Example Question #43 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

82950

10050

24750

13950

4050

Correct answer:

13950

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-4}^{1}\int_{5}^{10}\int_{-2}^{1}(6{(x + 6)}^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4}^{1}\int_{5}^{10}\int_{-2}^{1}(6{(x + 6)}^2)dxdydz=\int_{-4}^{1}\int_{5}^{10}(2{(x + 6)}^3)dydz|_{-2}^{1}\\&\int_{-4}^{1}\int_{5}^{10}(558)dydz=\int_{-4}^{1}(558y)dz|_{5}^{10}\\&\int_{-4}^{1}(2790)dz=2790zdz|_{-4}^{1}=13950\end{align*}$

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Example Question #44 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{4}\int_{3}^{4}\int_{5}^{8}(4{(y - 2)}^3)dxdydz\end{align*}$

Possible Answers:

6075

8050

3240

225

Correct answer:

225

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{4}\int_{3}^{4}\int_{5}^{8}(4{(y - 2)}^3)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{4}\int_{3}^{4}\int_{5}^{8}(4{(y - 2)}^3)dxdydz=\int_{-1}^{4}\int_{3}^{4}(4x{(y - 2)}^3)dydz|_{5}^{8}\\&\int_{-1}^{4}\int_{3}^{4}(12{(y - 2)}^3)dydz=\int_{-1}^{4}(3{(y - 2)}^4)dz|_{3}^{4}\\&\int_{-1}^{4}(45)dz=45zdz|_{-1}^{4}=225\end{align*}$

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Example Question #45 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{9}\int_{2}^{3}\int_{3}^{8}(4{(y - 7)}^3 - 4x)dxdydz\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{9}\int_{2}^{3}\int_{3}^{8}(4{(y - 7)}^3 - 4x)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{9}\int_{2}^{3}\int_{3}^{8}(4{(y - 7)}^3 - 4x)dxdydz=\int_{4}^{9}\int_{2}^{3}(4x{(y - 7)}^3 - 2x^2)dydz|_{3}^{8}\\&\int_{4}^{9}\int_{2}^{3}(20{(y - 7)}^3 - 110)dydz=\int_{4}^{9}(5{({(y - 7)}^3 - 22)}{(y - 7)})dz|_{2}^{3}\\&\int_{4}^{9}(-1955)dz=-1955zdz|_{4}^{9}=-9775\end{align*}$

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Example Question #46 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{2}\int_{5}^{6}\int_{2}^{4}(2x - 4z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{2}\int_{5}^{6}\int_{2}^{4}(2x - 4z)dxdydz=\int_{-1}^{2}\int_{5}^{6}(x{(x - 4z)})dydz|_{2}^{4}\\&\int_{-1}^{2}\int_{5}^{6}(12 - 8z)dydz=\int_{-1}^{2}(-y{(8z - 12)})dz|_{5}^{6}\\&\int_{-1}^{2}(12 - 8z)dz=-4z{(z - 3)}dz|_{-1}^{2}=24\end{align*}$

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Example Question #47 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{3}\int_{-3}^{0}\int_{3}^{7}(y^3 - x)dxdydz\end{align*}$

Possible Answers:

7032

6912

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{3}\int_{-3}^{0}\int_{3}^{7}(y^3 - x)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{3}\int_{-3}^{0}\int_{3}^{7}(y^3 - x)dxdydz=\int_{-1}^{3}\int_{-3}^{0}(-{(x{(x - 2y^3)})}/2)dydz|_{3}^{7}\\&\int_{-1}^{3}\int_{-3}^{0}(4y^3 - 20)dydz=\int_{-1}^{3}(y^4 - 20y)dz|_{-3}^{0}\\&\int_{-1}^{3}(-141)dz=-141zdz|_{-1}^{3}=-564\end{align*}$

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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:

1520/3

350/3

2600/3

520/3

560/3

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 #42 : 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 #183 : Multiple Integration

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

Possible Answers:

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

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

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

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

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

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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Calculus 3 : Triple Integration in Cartesian Coordinates

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #41 : Triple Integration In Cartesian Coordinates

Example Question #42 : Triple Integration In Cartesian Coordinates

Example Question #43 : Triple Integration In Cartesian Coordinates

Example Question #44 : Triple Integration In Cartesian Coordinates

Example Question #45 : Triple Integration In Cartesian Coordinates

Example Question #46 : Triple Integration In Cartesian Coordinates

Example Question #47 : Triple Integration In Cartesian Coordinates

Example Question #41 : Triple Integration In Cartesian Coordinates

Example Question #42 : Triple Integration In Cartesian Coordinates

Example Question #183 : Multiple Integration

All Calculus 3 Resources

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