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Calculus 3 : Multiple Integration

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Example Questions

Example Question #621 : Calculus 3

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

Possible Answers:

$\frac{117171}{14}$

$\frac{168795}{14}$

$\frac{84411}{14}$

$\frac{275895}{14}$

$\frac{117135}{14}$

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 #11 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

3366

3324

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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Example Question #151 : Multiple Integration

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

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 #152 : Triple Integrals

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

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 #153 : Triple Integrals

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

Possible Answers:

845

12675

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 #154 : Triple Integrals

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

Possible Answers:

2010

1470

1320

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 #155 : Triple Integrals

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

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 #156 : Triple Integrals

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

Possible Answers:

1035/2

11115

9005/2 5795/2

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 #21 : 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:

179160

1774680

1513560

1781280

178272

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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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #621 : Calculus 3

Example Question #151 : Multiple Integration

Example Question #11 : Triple Integration In Cartesian Coordinates

Example Question #151 : Multiple Integration

Example Question #152 : Triple Integrals

Example Question #153 : Triple Integrals

Example Question #154 : Triple Integrals

Example Question #155 : Triple Integrals

Example Question #156 : Triple Integrals

Example Question #21 : Triple Integration In Cartesian Coordinates

All Calculus 3 Resources

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