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Calculus 3 : Multiple Integration

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Example Questions

Example Question #22 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

2190

1510

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 #21 : 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:

1345

1015

19675

17515

11425

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 #23 : 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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Example Question #164 : Multiple Integration

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

Possible Answers:

600

500

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{5}^{10}\int_{-2}^{3}\int_{1}^{5}(2x + 4y - 3x^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{5}^{10}\int_{-2}^{3}\int_{1}^{5}(2x + 4y - 3x^2)dxdydz=\int_{5}^{10}\int_{-2}^{3}(x{(x + 4y - x^2)})dydz|_{1}^{5}\\&\int_{5}^{10}\int_{-2}^{3}(16y - 100)dydz=\int_{5}^{10}(4y{(2y - 25)})dz|_{-2}^{3}\\&\int_{5}^{10}(-460)dz=-460zdz|_{5}^{10}=-2300\end{align*}$

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Example Question #162 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{6}\int_{-4}^{0}\int_{0}^{2}(6x - 2y + 3y^2)dxdydz\end{align*}$

Possible Answers:

624

1128

696

1440

Correct answer:

624

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{3}^{6}\int_{-4}^{0}\int_{0}^{2}(6x - 2y + 3y^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{6}\int_{-4}^{0}\int_{0}^{2}(6x - 2y + 3y^2)dxdydz=\int_{3}^{6}\int_{-4}^{0}(3x^2 - x{(2y - 3y^2)})dydz|_{0}^{2}\\&\int_{3}^{6}\int_{-4}^{0}(6y^2 - 4y + 12)dydz=\int_{3}^{6}(2y{(y^2 - y + 6)})dz|_{-4}^{0}\\&\int_{3}^{6}(208)dz=208zdz|_{3}^{6}=624\end{align*}$

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Example Question #31 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{2}\int_{-4}^{-2}\int_{-5}^{-3}(5x + 2y - z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{2}\int_{-4}^{-2}\int_{-5}^{-3}(5x + 2y - z)dxdydz=\int_{-1}^{2}\int_{-4}^{-2}({(5x^2)}/2 + x{(2y - z)})dydz|_{-5}^{-3}\\&\int_{-1}^{2}\int_{-4}^{-2}(4y - 2z - 40)dydz=\int_{-1}^{2}(-2y{(z - y + 20)})dz|_{-4}^{-2}\\&\int_{-1}^{2}(- 4z - 104)dz=-2z{(z + 52)}dz|_{-1}^{2}=-318\end{align*}$

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Example Question #31 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{2}^{7}\int_{0}^{5}\int_{0}^{5}(2x + 6y^2 + 14)dxdydz\end{align*}$

Possible Answers:

9125

8625

19125

7045

Correct answer:

8625

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{7}\int_{0}^{5}\int_{0}^{5}(2x + 6y^2 + 14)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{7}\int_{0}^{5}\int_{0}^{5}(2x + 6y^2 + 14)dxdydz=\int_{2}^{7}\int_{0}^{5}(x{(6y^2 + 14)} + x^2)dydz|_{0}^{5}\\&\int_{2}^{7}\int_{0}^{5}(30y^2 + 95)dydz=\int_{2}^{7}(95y + 10y^3)dz|_{0}^{5}\\&\int_{2}^{7}(1725)dz=1725zdz|_{2}^{7}=8625\end{align*}$

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Example Question #168 : Multiple Integration

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

Possible Answers:

740

560

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{3}\int_{-2}^{3}\int_{-3}^{1}(14x - 3y^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{3}\int_{-2}^{3}\int_{-3}^{1}(14x - 3y^2)dxdydz=\int_{2}^{3}\int_{-2}^{3}(x{(7x - 3y^2)})dydz|_{-3}^{1}\\&\int_{2}^{3}\int_{-2}^{3}(- 12y^2 - 56)dydz=\int_{2}^{3}(-4y{(y^2 + 14)})dz|_{-2}^{3}\\&\int_{2}^{3}(-420)dz=-420zdz|_{2}^{3}=-420\end{align*}$

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Example Question #161 : Multiple Integration

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

Possible Answers:

$10e^{(7)} - 10e^{(2)} - 600$

$25e^{(-1)} - 25e^{(-3)} + 1350$

$25e^{(-1)} - 25e^{(-3)} + 150$

$10e^{(3)} - 10e^{(-2)} + 1350$

$10e^{(3)} - 10e^{(-2)} - 600$

Correct answer:

$25e^{(-1)} - 25e^{(-3)} + 1350$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-2}^{3}\int_{2}^{7}\int_{-3}^{-1}(6y + e^{(x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-2}^{3}\int_{2}^{7}\int_{-3}^{-1}(6y + e^{(x)})dxdydz=\int_{-2}^{3}\int_{2}^{7}(e^{(x)} + 6xy)dydz|_{-3}^{-1}\\&\int_{-2}^{3}\int_{2}^{7}(12y + e^{(-1)} - e^{(-3)})dydz=\int_{-2}^{3}(y{(e^{(-1)} - e^{(-3)})} + 6y^2)dz|_{2}^{7}\\&\int_{-2}^{3}(5e^{(-1)} - 5e^{(-3)} + 270)dz=z{(5e^{(-1)} - 5e^{(-3)} + 270)}dz|_{-2}^{3}=25e^{(-1)} - 25e^{(-3)} + 1350\end{align*}$

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Example Question #31 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

11sin{(3)} - 11sin{(2)} + 26

4sin{(3)} - 4sin{(2)} + 182

4sin{(6)} - 4sin{(5)} + 38

{(5sin{(3)})}/2 - {(5sin{(1)})}/2 + 182

5sin{(6)} - 5sin{(5)} + 26

Correct answer:

4sin{(3)} - 4sin{(2)} + 182

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{5}^{6}\int_{1}^{3}\int_{2}^{3}(ycos{(x)} + 3z^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{5}^{6}\int_{1}^{3}\int_{2}^{3}(ycos{(x)} + 3z^2)dxdydz=\int_{5}^{6}\int_{1}^{3}(3xz^2 + ysin{(x)})dydz|_{2}^{3}\\&\int_{5}^{6}\int_{1}^{3}(3z^2 - y{(sin{(2)} - sin{(3)})})dydz=\int_{5}^{6}(3yz^2 - {(y^2{(sin{(2)} - sin{(3)})})}/2)dz|_{1}^{3}\\&\int_{5}^{6}(4sin{(3)} - 4sin{(2)} + 6z^2)dz=2z^3 - z{(4sin{(2)} - 4sin{(3)})}dz|_{5}^{6}=4sin{(3)} - 4sin{(2)} + 182\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #22 : Triple Integration In Cartesian Coordinates

Example Question #21 : Triple Integration In Cartesian Coordinates

Example Question #23 : Triple Integration In Cartesian Coordinates

Example Question #164 : Multiple Integration

Example Question #162 : Triple Integrals

Example Question #31 : Triple Integration In Cartesian Coordinates

Example Question #31 : Triple Integration In Cartesian Coordinates

Example Question #168 : Multiple Integration

Example Question #161 : Multiple Integration

Example Question #31 : Triple Integration In Cartesian Coordinates

All Calculus 3 Resources

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