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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #142 : Multiple Integration

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

Possible Answers:

489216

23520

134064

222264

27048

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 #142 : Multiple Integration

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

Possible Answers:

4590

2820

3690

2010

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 #11 : 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

170

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 #12 : 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 #144 : Multiple Integration

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

Possible Answers:

1460

740

1000

100

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 #146 : Multiple Integration

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

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 #147 : Multiple Integration

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

Possible Answers:

16850

950

1750

350

1250

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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Example Question #151 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{7}\int_{-7}^{-2}\int_{-10}^{-7}(6z^2 - 1/y^2)dxdydz\end{align*}$

Possible Answers:

$\frac{117171}{14}$

$\frac{168795}{14}$

$\frac{117135}{14}$

$\frac{275895}{14}$

$\frac{84411}{14}$

Correct answer:

$\frac{117135}{14}$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{7}\int_{-7}^{-2}\int_{-10}^{-7}(6z^2 - 1/y^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{7}\int_{-7}^{-2}\int_{-10}^{-7}(6z^2 - 1/y^2)dxdydz=\int_{4}^{7}\int_{-7}^{-2}(-x{(1/y^2 - 6z^2)})dydz|_{-10}^{-7}\\&\int_{4}^{7}\int_{-7}^{-2}(18z^2 - 3/y^2)dydz=\int_{4}^{7}(18yz^2 + 3/y)dz|_{-7}^{-2}\\&\int_{4}^{7}(90z^2 - 15/14)dz=30z^3 - {(15z)}/14dz|_{4}^{7}=117135/14\end{align*}$

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Example Question #151 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{8}^{13}\int_{10}^{11}\int_{-2}^{0}(e^{(x + y)})dxdydz\end{align*}$

Possible Answers:

$2{(e^{(8)} - e^{(13)})}{(e^{(10)} - e^{(11)})}$

$3{(e^{(8)} - e^{(13)})}{(e^{(-2)} - 1)}$

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

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

${(e^{(8)} - e^{(13)})}{(e^{(-2)} - 1)}$

Correct answer:

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

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{8}^{13}\int_{10}^{11}\int_{-2}^{0}(e^{(x + y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{8}^{13}\int_{10}^{11}\int_{-2}^{0}(e^{(x + y)})dxdydz=\int_{8}^{13}\int_{10}^{11}(e^{(x + y)})dydz|_{-2}^{0}\\&\int_{8}^{13}\int_{10}^{11}(e^{(-2)}e^{(y)}{(e^{(2)} - 1)})dydz=\int_{8}^{13}(e^{(-2)}e^{(y)}{(e^{(2)} - 1)})dz|_{10}^{11}\\&\int_{8}^{13}(e^{(8)}{(e^{(1)} - 1)}^2{(e^{(1)} + 1)})dz=ze^{(8)}{(e^{(1)} - 1)}^2{(e^{(1)} + 1)}dz|_{8}^{13}=5e^{(8)}{(e^{(1)} - 1)}^2{(e^{(1)} + 1)}\end{align*}$

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Example Question #153 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-6}\int_{4}^{5}\int_{-1}^{0}(12z^3 - 2x)dxdydz\end{align*}$

Possible Answers:

3324

3366

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-9}^{-6}\int_{4}^{5}\int_{-1}^{0}(12z^3 - 2x)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-6}\int_{4}^{5}\int_{-1}^{0}(12z^3 - 2x)dxdydz=\int_{-9}^{-6}\int_{4}^{5}(12xz^3 - x^2)dydz|_{-1}^{0}\\&\int_{-9}^{-6}\int_{4}^{5}(12z^3 + 1)dydz=\int_{-9}^{-6}(y{(12z^3 + 1)})dz|_{4}^{5}\\&\int_{-9}^{-6}(12z^3 + 1)dz=z + 3z^4dz|_{-9}^{-6}=-15792\end{align*}$

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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 #142 : Multiple Integration

Example Question #142 : Multiple Integration

Example Question #11 : Triple Integration In Cartesian Coordinates

Example Question #12 : Triple Integration In Cartesian Coordinates

Example Question #144 : Multiple Integration

Example Question #146 : Multiple Integration

Example Question #147 : Multiple Integration

Example Question #151 : Multiple Integration

Example Question #151 : Multiple Integration

Example Question #153 : Multiple Integration

All Calculus 3 Resources

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