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Calculus 3 : Multiple Integration

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Example Questions

Example Question #1 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-13}^{2}\int_{-13}^{-7}\int_{6}^{9}(3x + y + 2z)dxdydz\end{align*}$

Possible Answers:

405

Correct answer:

405

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-13}^{2}\int_{-13}^{-7}\int_{6}^{9}(3x + y + 2z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-13}^{2}\int_{-13}^{-7}\int_{6}^{9}(3x + y + 2z)dxdydz=\int_{-13}^{2}\int_{-13}^{-7}(x{(y + 2z)} + {(3x^2)}/2)dydz|_{6}^{9}\\&\int_{-13}^{2}\int_{-13}^{-7}(3y + 6z + 135/2)dydz=\int_{-13}^{2}({(3y{(y + 4z + 45)})}/2)dz|_{-13}^{-7}\\&\int_{-13}^{2}(36z + 225)dz=9z{(2z + 25)}dz|_{-13}^{2}=405\end{align*}$

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Example Question #1 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{8}^{20}\int_{-7}^{-2}\int_{-11}^{18}(6xy - 2z)dxdydz\end{align*}$

Possible Answers:

527220

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{8}^{20}\int_{-7}^{-2}\int_{-11}^{18}(6xy - 2z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{8}^{20}\int_{-7}^{-2}\int_{-11}^{18}(6xy - 2z)dxdydz=\int_{8}^{20}\int_{-7}^{-2}(3x^2y - 2xz)dydz|_{-11}^{18}\\&\int_{8}^{20}\int_{-7}^{-2}(609y - 58z)dydz=\int_{8}^{20}({(29y{(21y - 4z)})}/2)dz|_{-7}^{-2}\\&\int_{8}^{20}(- 290z - 27405/2)dz=-{(145z{(2z + 189)})}/2dz|_{8}^{20}=-213150\end{align*}$

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Example Question #142 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-4}^{3}\int_{-8}^{-2}\int_{4}^{6}(24xyz^2)dxdydz\end{align*}$

Possible Answers:

127680

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-4}^{3}\int_{-8}^{-2}\int_{4}^{6}(24xyz^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4}^{3}\int_{-8}^{-2}\int_{4}^{6}(24xyz^2)dxdydz=\int_{-4}^{3}\int_{-8}^{-2}(12x^2yz^2)dydz|_{4}^{6}\\&\int_{-4}^{3}\int_{-8}^{-2}(240yz^2)dydz=\int_{-4}^{3}(120y^2z^2)dz|_{-8}^{-2}\\&\int_{-4}^{3}(-7200z^2)dz=-2400z^3dz|_{-4}^{3}=-218400\end{align*}$

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Example Question #11 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-10}^{11}\int_{-5}^{9}\int_{0}^{2}(24xy^2 - 8z^3)dxdydz\end{align*}$

Possible Answers:

23520

222264

27048

489216

134064

Correct answer:

27048

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-10}^{11}\int_{-5}^{9}\int_{0}^{2}(24xy^2 - 8z^3)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-10}^{11}\int_{-5}^{9}\int_{0}^{2}(24xy^2 - 8z^3)dxdydz=\int_{-10}^{11}\int_{-5}^{9}(12x^2y^2 - 8xz^3)dydz|_{0}^{2}\\&\int_{-10}^{11}\int_{-5}^{9}(48y^2 - 16z^3)dydz=\int_{-10}^{11}(16y{(y^2 - z^3)})dz|_{-5}^{9}\\&\int_{-10}^{11}(13664 - 224z^3)dz=-56z{(z^3 - 244)}dz|_{-10}^{11}=27048\end{align*}$

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Example Question #12 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{4}\int_{2}^{4}\int_{5}^{8}(12x - 8z + 3y^2)dxdydz\end{align*}$

Possible Answers:

4590

2820

2010

3690

Correct answer:

2820

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{4}\int_{2}^{4}\int_{5}^{8}(12x - 8z + 3y^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{4}\int_{2}^{4}\int_{5}^{8}(12x - 8z + 3y^2)dxdydz=\int_{-1}^{4}\int_{2}^{4}(6x^2 - x{(8z - 3y^2)})dydz|_{5}^{8}\\&\int_{-1}^{4}\int_{2}^{4}(9y^2 - 24z + 234)dydz=\int_{-1}^{4}(3y^3 - y{(24z - 234)})dz|_{2}^{4}\\&\int_{-1}^{4}(636 - 48z)dz=-12z{(2z - 53)}dz|_{-1}^{4}=2820\end{align*}$

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Example Question #13 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

170

226

Correct answer:

170

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{5}\int_{-5}^{-3}\int_{6}^{7}(2x - 4yz)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{5}\int_{-5}^{-3}\int_{6}^{7}(2x - 4yz)dxdydz=\int_{4}^{5}\int_{-5}^{-3}(x^2 - 4xyz)dydz|_{6}^{7}\\&\int_{4}^{5}\int_{-5}^{-3}(13 - 4yz)dydz=\int_{4}^{5}(-y{(2yz - 13)})dz|_{-5}^{-3}\\&\int_{4}^{5}(32z + 26)dz=2z{(8z + 13)}dz|_{4}^{5}=170\end{align*}$

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Example Question #14 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{10}\int_{3}^{6}\int_{5}^{6}(4x - 12y)dxdydz\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{6}^{10}\int_{3}^{6}\int_{5}^{6}(4x - 12y)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{10}\int_{3}^{6}\int_{5}^{6}(4x - 12y)dxdydz=\int_{6}^{10}\int_{3}^{6}(2x{(x - 6y)})dydz|_{5}^{6}\\&\int_{6}^{10}\int_{3}^{6}(22 - 12y)dydz=\int_{6}^{10}(-2y{(3y - 11)})dz|_{3}^{6}\\&\int_{6}^{10}(-96)dz=-96zdz|_{6}^{10}=-384\end{align*}$

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Example Question #15 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-5}\int_{-3}^{1}\int_{8}^{13}(2x + 4y - 6z)dxdydz\end{align*}$

Possible Answers:

740

1460

100

1000

Correct answer:

1000

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-6}^{-5}\int_{-3}^{1}\int_{8}^{13}(2x + 4y - 6z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-5}\int_{-3}^{1}\int_{8}^{13}(2x + 4y - 6z)dxdydz=\int_{-6}^{-5}\int_{-3}^{1}(x^2 + x{(4y - 6z)})dydz|_{8}^{13}\\&\int_{-6}^{-5}\int_{-3}^{1}(20y - 30z + 105)dydz=\int_{-6}^{-5}(10y^2 - y{(30z - 105)})dz|_{-3}^{1}\\&\int_{-6}^{-5}(340 - 120z)dz=-20z{(3z - 17)}dz|_{-6}^{-5}=1000\end{align*}$

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Example Question #16 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{2}^{6}\int_{0}^{5}\int_{5}^{8}(5x^4 - 12y^5)dxdydz\end{align*}$

Possible Answers:

217860

Correct answer:

217860

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{6}\int_{0}^{5}\int_{5}^{8}(5x^4 - 12y^5)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{6}\int_{0}^{5}\int_{5}^{8}(5x^4 - 12y^5)dxdydz=\int_{2}^{6}\int_{0}^{5}(x^5 - 12xy^5)dydz|_{5}^{8}\\&\int_{2}^{6}\int_{0}^{5}(29643 - 36y^5)dydz=\int_{2}^{6}(29643y - 6y^6)dz|_{0}^{5}\\&\int_{2}^{6}(54465)dz=54465zdz|_{2}^{6}=217860\end{align*}$

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Example Question #141 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-5}^{0}\int_{2}^{3}\int_{8}^{13}(6z^2)dxdydz\end{align*}$

Possible Answers:

16850

350

1250

1750

950

Correct answer:

1250

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-5}^{0}\int_{2}^{3}\int_{8}^{13}(6z^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-5}^{0}\int_{2}^{3}\int_{8}^{13}(6z^2)dxdydz=\int_{-5}^{0}\int_{2}^{3}(6xz^2)dydz|_{8}^{13}\\&\int_{-5}^{0}\int_{2}^{3}(30z^2)dydz=\int_{-5}^{0}(30yz^2)dz|_{2}^{3}\\&\int_{-5}^{0}(30z^2)dz=10z^3dz|_{-5}^{0}=1250\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1 : Triple Integration In Cartesian Coordinates

Example Question #1 : Triple Integration In Cartesian Coordinates

Example Question #142 : Multiple Integration

Example Question #11 : Triple Integration In Cartesian Coordinates

Example Question #12 : Triple Integration In Cartesian Coordinates

Example Question #13 : Triple Integration In Cartesian Coordinates

Example Question #14 : Triple Integration In Cartesian Coordinates

Example Question #15 : Triple Integration In Cartesian Coordinates

Example Question #16 : Triple Integration In Cartesian Coordinates

Example Question #141 : Triple Integrals

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