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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #21 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-3}\int_{7}^{11}\int_{-8}^{-3}(3x^2 + 16z^3)dxdydz\end{align*}$

Possible Answers:

986960

985520

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-7}^{-3}\int_{7}^{11}\int_{-8}^{-3}(3x^2 + 16z^3)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-3}\int_{7}^{11}\int_{-8}^{-3}(3x^2 + 16z^3)dxdydz=\int_{-7}^{-3}\int_{7}^{11}(16xz^3 + x^3)dydz|_{-8}^{-3}\\&\int_{-7}^{-3}\int_{7}^{11}(80z^3 + 485)dydz=\int_{-7}^{-3}(y{(80z^3 + 485)})dz|_{7}^{11}\\&\int_{-7}^{-3}(320z^3 + 1940)dz=20z{(4z^3 + 97)}dz|_{-7}^{-3}=-177840\end{align*}$

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Example Question #22 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{2}\int_{3}^{8}\int_{0}^{5}(3z^2 - 6y^2)dxdydz\end{align*}$

Possible Answers:

6825

3525

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{2}\int_{3}^{8}\int_{0}^{5}(3z^2 - 6y^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{2}\int_{3}^{8}\int_{0}^{5}(3z^2 - 6y^2)dxdydz=\int_{-1}^{2}\int_{3}^{8}(-x{(6y^2 - 3z^2)})dydz|_{0}^{5}\\&\int_{-1}^{2}\int_{3}^{8}(15z^2 - 30y^2)dydz=\int_{-1}^{2}(15yz^2 - 10y^3)dz|_{3}^{8}\\&\int_{-1}^{2}(75z^2 - 4850)dz=25z{(z^2 - 194)}dz|_{-1}^{2}=-14325\end{align*}$

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Example Question #23 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

12675

845

Correct answer:

845

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-2}^{3}\int_{-8}^{-7}\int_{7}^{8}(6y^2z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-2}^{3}\int_{-8}^{-7}\int_{7}^{8}(6y^2z)dxdydz=\int_{-2}^{3}\int_{-8}^{-7}(6xy^2z)dydz|_{7}^{8}\\&\int_{-2}^{3}\int_{-8}^{-7}(6y^2z)dydz=\int_{-2}^{3}(2y^3z)dz|_{-8}^{-7}\\&\int_{-2}^{3}(338z)dz=169z^2dz|_{-2}^{3}=845\end{align*}$

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Example Question #24 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-2}^{3}\int_{-2}^{1}\int_{6}^{7}(14y - 6x^2z)dxdydz\end{align*}$

Possible Answers:

1470

1320

2010

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-2}^{3}\int_{-2}^{1}\int_{6}^{7}(14y - 6x^2z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-2}^{3}\int_{-2}^{1}\int_{6}^{7}(14y - 6x^2z)dxdydz=\int_{-2}^{3}\int_{-2}^{1}(14xy - 2x^3z)dydz|_{6}^{7}\\&\int_{-2}^{3}\int_{-2}^{1}(14y - 254z)dydz=\int_{-2}^{3}(y{(7y - 254z)})dz|_{-2}^{1}\\&\int_{-2}^{3}(- 762z - 21)dz=-3z{(127z + 7)}dz|_{-2}^{3}=-2010\end{align*}$

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Example Question #25 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{11}\int_{-1}^{0}\int_{3}^{5}(4x - 2y + 4z)dxdydz\end{align*}$

Possible Answers:

118

Correct answer:

118

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{10}^{11}\int_{-1}^{0}\int_{3}^{5}(4x - 2y + 4z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{11}\int_{-1}^{0}\int_{3}^{5}(4x - 2y + 4z)dxdydz=\int_{10}^{11}\int_{-1}^{0}(2x^2 - x{(2y - 4z)})dydz|_{3}^{5}\\&\int_{10}^{11}\int_{-1}^{0}(8z - 4y + 32)dydz=\int_{10}^{11}(y{(8z + 32)} - 2y^2)dz|_{-1}^{0}\\&\int_{10}^{11}(8z + 34)dz=2z{(2z + 17)}dz|_{10}^{11}=118\end{align*}$

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Example Question #26 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

9005/2 5795/2

11115

1035/2

2885/2

Correct answer:

1035/2

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-3}^{2}\int_{-6}^{-3}\int_{-8}^{-6}(x + y^2 - z^3)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{2}\int_{-6}^{-3}\int_{-8}^{-6}(x + y^2 - z^3)dxdydz=\int_{-3}^{2}\int_{-6}^{-3}(x{(y^2 - z^3)} + x^2/2)dydz|_{-8}^{-6}\\&\int_{-3}^{2}\int_{-6}^{-3}(2y^2 - 2z^3 - 14)dydz=\int_{-3}^{2}({(2y^3)}/3 - y{(2z^3 + 14)})dz|_{-6}^{-3}\\&\int_{-3}^{2}(84 - 6z^3)dz=-{(3z{(z^3 - 56)})}/2dz|_{-3}^{2}=1035/2\end{align*}$

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Example Question #27 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{8}\int_{10}^{12}\int_{9}^{12}(5z^4 - 3{(x + 2)}^2)dxdydz\end{align*}$

Possible Answers:

1781280

1513560

178272

1774680

179160

Correct answer:

179160

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{8}\int_{10}^{12}\int_{9}^{12}(5z^4 - 3{(x + 2)}^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{8}\int_{10}^{12}\int_{9}^{12}(5z^4 - 3{(x + 2)}^2)dxdydz=\int_{4}^{8}\int_{10}^{12}(5xz^4 - {(x + 2)}^3)dydz|_{9}^{12}\\&\int_{4}^{8}\int_{10}^{12}(15z^4 - 1413)dydz=\int_{4}^{8}(y{(15z^4 - 1413)})dz|_{10}^{12}\\&\int_{4}^{8}(30z^4 - 2826)dz=6z{(z^4 - 471)}dz|_{4}^{8}=179160\end{align*}$

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Example Question #161 : Multiple Integration

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

Possible Answers:

1510

2190

1280

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-8}^{-3}\int_{3}^{4}\int_{1}^{5}(2z^3 - 4y)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-8}^{-3}\int_{3}^{4}\int_{1}^{5}(2z^3 - 4y)dxdydz=\int_{-8}^{-3}\int_{3}^{4}(-x{(4y - 2z^3)})dydz|_{1}^{5}\\&\int_{-8}^{-3}\int_{3}^{4}(8z^3 - 16y)dydz=\int_{-8}^{-3}(-8y{(y - z^3)})dz|_{3}^{4}\\&\int_{-8}^{-3}(8z^3 - 56)dz=2z{(z^3 - 28)}dz|_{-8}^{-3}=-8310\end{align*}$

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Example Question #28 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{5}\int_{1}^{6}\int_{-7}^{-6}(24{(y + 2z)}^2 + 3x^2)dxdydz\end{align*}$

Possible Answers:

1015

1345

19675

11425

17515

Correct answer:

19675

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{5}\int_{1}^{6}\int_{-7}^{-6}(24{(y + 2z)}^2 + 3x^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{5}\int_{1}^{6}\int_{-7}^{-6}(24{(y + 2z)}^2 + 3x^2)dxdydz=\int_{4}^{5}\int_{1}^{6}(x^3 + 24x{(y + 2z)}^2)dydz|_{-7}^{-6}\\&\int_{4}^{5}\int_{1}^{6}(24{(y + 2z)}^2 + 127)dydz=\int_{4}^{5}(127y + 8{(y + 2z)}^3)dz|_{1}^{6}\\&\int_{4}^{5}(1680z + 480z^2 + 2355)dz=5z{(168z + 32z^2 + 471)}dz|_{4}^{5}=19675\end{align*}$

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Example Question #29 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{11}\int_{10}^{11}\int_{-3}^{0}(8xz^3)dxdydz\end{align*}$

Possible Answers:

771120

1002456

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{7}^{11}\int_{10}^{11}\int_{-3}^{0}(8xz^3)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{11}\int_{10}^{11}\int_{-3}^{0}(8xz^3)dxdydz=\int_{7}^{11}\int_{10}^{11}(4x^2z^3)dydz|_{-3}^{0}\\&\int_{7}^{11}\int_{10}^{11}(-36z^3)dydz=\int_{7}^{11}(-36yz^3)dz|_{10}^{11}\\&\int_{7}^{11}(-36z^3)dz=-9z^4dz|_{7}^{11}=-110160\end{align*}$

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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 #21 : Triple Integration In Cartesian Coordinates

Example Question #22 : Triple Integration In Cartesian Coordinates

Example Question #23 : Triple Integration In Cartesian Coordinates

Example Question #24 : Triple Integration In Cartesian Coordinates

Example Question #25 : Triple Integration In Cartesian Coordinates

Example Question #26 : Triple Integration In Cartesian Coordinates

Example Question #27 : Triple Integration In Cartesian Coordinates

Example Question #161 : Multiple Integration

Example Question #28 : Triple Integration In Cartesian Coordinates

Example Question #29 : Triple Integration In Cartesian Coordinates

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