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Calculus 3 : Multiple Integration

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Example Questions

Example Question #141 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{0.5}\int_{6}^{7.5}\int_{8}^{9}(\frac{(8x^{2}y^{2}e^{(-2z)})}{69})dxdydz\end{align*}$

Possible Answers:

116000

232000

Correct answer:

116000

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral, and keeping savvy about }\\&\text{utilizing integral rules:}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\end{align*}$

$\begin{align*}&\int_{-3}^{0.5}\int_{6}^{7.5}\int_{8}^{9}(\frac{(8x^{2}y^{2}e^{(-2z)})}{69})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{0.5}\int_{6}^{7.5}\int_{8}^{9}(\frac{(8x^{2}y^{2}e^{(-2z)})}{69})dxdydz=\int_{-3}^{0.5}\int_{6}^{7.5}(\frac{(8x^{3}y^{2}e^{(-2z)})}{207})dydz|_{8}^{9}\\&\int_{-3}^{0.5}\int_{6}^{7.5}(\frac{(1736y^{2}e^{(-2z)})}{207})dydz=\int_{-3}^{0.5}(\frac{(1736y^{3}e^{(-2z)})}{621})dz|_{6}^{7.5}\\&\int_{-3}^{0.5}(\frac{(13237e^{(-2z)})}{23})dz=-\frac{(13237e^{(-2z)})}{46}|_{-3}^{0.5}=116000\end{align*}$

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Example Question #282 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{3.5}^{7.5}\int_{-6}^{-4.5}\int_{9}^{13}(34xcos(3y)e^{(-2z)})dxdydz\end{align*}$

Possible Answers:

-0.35

0.07

2.12

-0.09

Correct answer:

-0.35

Explanation:

$\begin{align*}&\int_{3.5}^{7.5}\int_{-6}^{-4.5}\int_{9}^{13}(34xcos(3y)e^{(-2z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3.5}^{7.5}\int_{-6}^{-4.5}\int_{9}^{13}(34xcos(3y)e^{(-2z)})dxdydz=\int_{3.5}^{7.5}\int_{-6}^{-4.5}(17x^{2}cos(3y)e^{(-2z)})dydz|_{9}^{13}\\&\int_{3.5}^{7.5}\int_{-6}^{-4.5}(1496cos(3y)e^{(-2z)})dydz=\int_{3.5}^{7.5}(\frac{(1496sin(3y)e^{(-2z)})}{3})dz|_{-6}^{-4.5}\\&\int_{3.5}^{7.5}(\frac{(1496e^{(-2z)}\cdot(sin(18) - sin(\frac{27}{2})))}{3})dz=-\frac{(748e^{(-2z)}\cdot(sin(18) - sin(\frac{27}{2})))}{3}|_{3.5}^{7.5}=-0.35\end{align*}$

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Example Question #281 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-5}^{-2}\int_{-4}^{1}\int_{9}^{12.5}(\frac{(11e^{(-2y)}e^{(z)})}{28})dxdydz\end{align*}$

Possible Answers:

1581

-52.71

263.5

-790.6

Correct answer:

263.5

Explanation:

$\begin{align*}&\int_{-5}^{-2}\int_{-4}^{1}\int_{9}^{12.5}(\frac{(11e^{(-2y)}e^{(z)})}{28})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-5}^{-2}\int_{-4}^{1}\int_{9}^{12.5}(\frac{(11e^{(-2y)}e^{(z)})}{28})dxdydz=\int_{-5}^{-2}\int_{-4}^{1}(\frac{(11xe^{(-2y)}e^{(z)})}{28})dydz|_{9}^{12.5}\\&\int_{-5}^{-2}\int_{-4}^{1}(\frac{(11e^{(z - 2y)})}{8})dydz=\int_{-5}^{-2}(-\frac{(11e^{(z - 2y)})}{16})dz|_{-4}^{1}\\&\int_{-5}^{-2}(\frac{(11e^{(-2)}e^{(z)}\cdot(e^{(10)} - 1))}{16})dz=e^{(z - 2)}\cdot(\frac{(11e^{(10)})}{16}-\frac{ 11}{16})|_{-5}^{-2}=263.5\end{align*}$

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Example Question #284 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{7.5}\int_{4.5}^{8.5}\int_{-5}^{-0.5}(\frac{(11z^{2}e^{(-x)}e^{(-y)})}{38})dxdydz\end{align*}$

Possible Answers:

-7.95

31.8

5.3

-190.8

Correct answer:

31.8

Explanation:

$\begin{align*}&\int_{6}^{7.5}\int_{4.5}^{8.5}\int_{-5}^{-0.5}(\frac{(11z^{2}e^{(-x)}e^{(-y)})}{38})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{7.5}\int_{4.5}^{8.5}\int_{-5}^{-0.5}(\frac{(11z^{2}e^{(-x)}e^{(-y)})}{38})dxdydz=\int_{6}^{7.5}\int_{4.5}^{8.5}(-\frac{(11z^{2}e^{(- x - y)})}{38})dydz|_{-5}^{-0.5}\\&\int_{6}^{7.5}\int_{4.5}^{8.5}(-\frac{(11z^{2}e^{(-y)}\cdot(e^{(\frac{1}{2})} - e^{(5)}))}{38})dydz=\int_{6}^{7.5}(\frac{(11z^{2}e^{(-y)}\cdot(e^{(\frac{1}{2})} - e^{(5)}))}{38})dz|_{4.5}^{8.5}\\&\int_{6}^{7.5}(-\frac{(11z^{2}\cdot(e^{(-8)} - e^{(-\frac{7}{2})})\cdot(e^{(4)} - 1))}{38})dz=z^{3}\cdot(\frac{(11e^{(\frac{1}{2})})}{114}-\frac{ (11e^{(-4)})}{114}+\frac{ (11e^{(-8)})}{114}-\frac{ (11e^{(-\frac{7}{2})})}{114})|_{6}^{7.5}=31.8\end{align*}$

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Example Question #285 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-5}\int_{-8}^{-4.5}\int_{-4}^{-2}(\frac{(4cos(3y)e^{(-2x)})}{z^{2}})dxdydz\end{align*}$

Possible Answers:

-111.2

-333.5

55.58

555.8

Correct answer:

-111.2

Explanation:

$\begin{align*}&\int_{-6}^{-5}\int_{-8}^{-4.5}\int_{-4}^{-2}(\frac{(4cos(3y)e^{(-2x)})}{z^{2}})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-5}\int_{-8}^{-4.5}\int_{-4}^{-2}(\frac{(4cos(3y)e^{(-2x)})}{z^{2}})dxdydz=\int_{-6}^{-5}\int_{-8}^{-4.5}(-\frac{(2cos(3y)e^{(-2x)})}{z^{2}})dydz|_{-4}^{-2}\\&\int_{-6}^{-5}\int_{-8}^{-4.5}(\frac{(2cos(3y)e^{(4)}\cdot(e^{(4)} - 1))}{z^{2}})dydz=\int_{-6}^{-5}(\frac{(2sin(3y)e^{(4)}\cdot(e^{(4)} - 1))}{(3z^{2})})dz|_{-8}^{-4.5}\\&\int_{-6}^{-5}(\frac{(2e^{(4)}\cdot(sin(24) - sin(\frac{27}{2}))\cdot(e^{(4)} - 1))}{(3z^{2})})dz=-\frac{(2e^{(4)}\cdot(sin(24) - sin(\frac{27}{2}))\cdot(e^{(4)} - 1))}{(3z)}|_{-6}^{-5}=-111.2\end{align*}$

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Example Question #286 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-1.5}\int_{-4}^{-2}\int_{4}^{8}(\frac{(13\cdot2^{(\frac{x}{3})}\cdot3^y)}{(5z^{2})})dxdydz\end{align*}$

Possible Answers:

11.62

-0.32

-11.62

1.94

Correct answer:

1.94

Explanation:

$\begin{align*}&\int_{-6}^{-1.5}\int_{-4}^{-2}\int_{4}^{8}(\frac{(13\cdot2^{(\frac{x}{3})}\cdot3^y)}{(5z^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-1.5}\int_{-4}^{-2}\int_{4}^{8}(\frac{(13\cdot2^{(\frac{x}{3})}\cdot3^y)}{(5z^{2})})dxdydz=\int_{-6}^{-1.5}\int_{-4}^{-2}(\frac{(39\cdot2^{(\frac{x}{3})}\cdot3^y)}{(5z^{2}ln(2))})dydz|_{4}^{8}\\&\int_{-6}^{-1.5}\int_{-4}^{-2}(-\frac{(13\cdot3^y\cdot(6\cdot2^{(\frac{1}{3})} - 12\cdot2^{(\frac{2}{3})}))}{(5z^{2}ln(2))})dydz=\int_{-6}^{-1.5}(-\frac{(78\cdot3^y\cdot(2^{(\frac{1}{3})} - 2\cdot2^{(\frac{2}{3})}))}{(5z^{2}ln(2)ln(3))})dz|_{-4}^{-2}\\&\int_{-6}^{-1.5}(-\frac{(208\cdot2^{(\frac{1}{3})} - 416\cdot2^{(\frac{2}{3})})}{(135z^{2}ln(2)ln(3))})dz=\frac{(208\cdot2^{(\frac{1}{3})} - 416\cdot4^{(\frac{1}{3})})}{(135zln(2)ln(3))}|_{-6}^{-1.5}=1.94\end{align*}$

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Example Question #287 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{8}^{13}\int_{10}^{12}\int_{-4}^{-2.5}(\frac{(z^{2}cos(3y))}{(7\cdot3^x)})dxdydz\end{align*}$

Possible Answers:

-1.19

5.97

-5.97

1.99

Correct answer:

-5.97

Explanation:

$\begin{align*}&\int_{8}^{13}\int_{10}^{12}\int_{-4}^{-2.5}(\frac{(z^{2}cos(3y))}{(7\cdot3^x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{8}^{13}\int_{10}^{12}\int_{-4}^{-2.5}(\frac{(z^{2}cos(3y))}{(7\cdot3^x)})dxdydz=\int_{8}^{13}\int_{10}^{12}(-\frac{(z^{2}cos(3y))}{(7\cdot3^xln(3))})dydz|_{-4}^{-2.5}\\&\int_{8}^{13}\int_{10}^{12}(-\frac{(9z^{2}cos(3y)\cdot(3^{(\frac{1}{2})} - 9))}{(7ln(3))})dydz=\int_{8}^{13}(-\frac{(3z^{2}sin(3y)\cdot(3^{(\frac{1}{2})} - 9))}{(7ln(3))})dz|_{10}^{12}\\&\int_{8}^{13}(\frac{(3z^{2}\cdot(sin(30) - sin(36))\cdot(3^{(\frac{1}{2})} - 9))}{(7ln(3))})dz=-\frac{(z^{3}\cdot(27sin(30) - 27sin(36) - 3\cdot3^{(\frac{1}{2})}sin(30) + 3\cdot3^{(\frac{1}{2})}sin(36)))}{(21ln(3))}|_{8}^{13}=-5.97\end{align*}$

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Example Question #288 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{15}\int_{-10}^{-7}\int_{8}^{9}(\frac{(23cos(z + 2)sin(3y))}{(18x)})dxdydz\end{align*}$

Possible Answers:

-0.01

0.12

-0.06

0.09

Correct answer:

-0.01

Explanation:

$\begin{align*}&\int_{10}^{15}\int_{-10}^{-7}\int_{8}^{9}(\frac{(23cos(z + 2)sin(3y))}{(18x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{15}\int_{-10}^{-7}\int_{8}^{9}(\frac{(23cos(z + 2)sin(3y))}{(18x)})dxdydz=\int_{10}^{15}\int_{-10}^{-7}(\frac{(23cos(z + 2)sin(3y)ln(x))}{18})dydz|_{8}^{9}\\&\int_{10}^{15}\int_{-10}^{-7}(\frac{(23cos(z + 2)sin(3y)ln(\frac{9}{8}))}{18})dydz=\int_{10}^{15}(-\frac{(23cos(3y)cos(z + 2)ln(\frac{9}{8}))}{54})dz|_{-10}^{-7}\\&\int_{10}^{15}(-\frac{(23cos(z + 2)ln(\frac{9}{8})\cdot(\frac{cos(21)}{3}-\frac{ cos(30)}{3}))}{18})dz=-\frac{(23sin(z + 2)ln(\frac{9}{8})\cdot(\frac{cos(21)}{3}-\frac{ cos(30)}{3}))}{18}|_{10}^{15}=-0.01\end{align*}$

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Example Question #289 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{-0.5}\int_{-4.5}^{-1.5}\int_{5}^{10}(\frac{(21\cdot2^{(\frac{x}{4})}\cdot3^{(\frac{y}{2})}\cdot3^{(\frac{z}{2})})}{2})dxdydz\end{align*}$

Possible Answers:

-26.47

132.3

264.7

Correct answer:

132.3

Explanation:

$\begin{align*}&\int_{-3}^{-0.5}\int_{-4.5}^{-1.5}\int_{5}^{10}(\frac{(21\cdot2^{(\frac{x}{4})}\cdot3^{(\frac{y}{2})}\cdot3^{(\frac{z}{2})})}{2})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{-0.5}\int_{-4.5}^{-1.5}\int_{5}^{10}(\frac{(21\cdot2^{(\frac{x}{4})}\cdot3^{(\frac{y}{2})}\cdot3^{(\frac{z}{2})})}{2})dxdydz=\int_{-3}^{-0.5}\int_{-4.5}^{-1.5}(\frac{(42\cdot2^{(\frac{x}{4})}\cdot3^{(\frac{y}{2}+\frac{ z}{2})})}{ln(2)})dydz|_{5}^{10}\end{align*}$

$\begin{align*}\\&\int_{-3}^{-0.5}\int_{-4.5}^{-1.5}(\frac{(84\cdot2^{(\frac{1}{4})}\cdot3^{(\frac{y}{2})}\cdot3^{(\frac{z}{2})}\cdot(2\cdot2^{(\frac{1}{4})} - 1))}{ln(2)})dydz=\int_{-3}^{-0.5}(\frac{(2\cdot(168\cdot2^{(\frac{1}{2})}\cdot3^{(\frac{y}{2})}\cdot3^{(\frac{z}{2})} - 84\cdot2^{(\frac{1}{4})}\cdot3^{(\frac{y}{2})}\cdot3^{(\frac{z}{2})}))}{(ln(2)ln(3))})dz|_{-4.5}^{-1.5}\\&\int_{-3}^{-0.5}(-\frac{(56\cdot2^{(\frac{1}{4})}\cdot3^{(\frac{1}{4})}\cdot3^{(\frac{z}{2})}\cdot(2\cdot2^{(\frac{1}{4})} - 1)\cdot(3^{(\frac{1}{2})} - 9))}{(9ln(2)ln(3))})dz=\frac{(2\cdot3^{(\frac{z}{2})}\cdot(1008\cdot2^{(\frac{1}{2})}\cdot3^{(\frac{1}{4})} - 112\cdot2^{(\frac{1}{2})}\cdot27^{(\frac{1}{4})} - 504\cdot6^{(\frac{1}{4})} + 56\cdot54^{(\frac{1}{4})}))}{(9ln(2)ln(3)^{2})}|_{-3}^{-0.5}=132.3\end{align*}$

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Example Question #290 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-4}\int_{-10}^{-7.5}\int_{4}^{8.5}(\frac{(cos(3y)cos(z + 1))}{(2\cdot2^{(\frac{x}{2})})})dxdydz\end{align*}$

Possible Answers:

0.05

-0.01

0.01

-0.31

Correct answer:

0.05

Explanation:

$\begin{align*}&\int_{-6}^{-4}\int_{-10}^{-7.5}\int_{4}^{8.5}(\frac{(cos(3y)cos(z + 1))}{(2\cdot2^{(\frac{x}{2})})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-4}\int_{-10}^{-7.5}\int_{4}^{8.5}(\frac{(cos(3y)cos(z + 1))}{(2\cdot2^{(\frac{x}{2})})})dxdydz=\int_{-6}^{-4}\int_{-10}^{-7.5}(-\frac{(cos(3y)cos(z + 1))}{(2^{(\frac{x}{2})}ln(2))})dydz|_{4}^{8.5}\\&\int_{-6}^{-4}\int_{-10}^{-7.5}(-\frac{(cos(3y)cos(z + 1)\cdot(2^{(\frac{3}{4})} - 8))}{(32ln(2))})dydz=\int_{-6}^{-4}(-\frac{(cos(z + 1)sin(3y)\cdot(2^{(\frac{3}{4})} - 8))}{(96ln(2))})dz|_{-10}^{-7.5}\\&\int_{-6}^{-4}(-\frac{(cos(z + 1)\cdot(sin(30) - sin(\frac{45}{2}))\cdot(2^{(\frac{3}{4})} - 8))}{(96ln(2))})dz=-\frac{(sin(z + 1)\cdot(sin(30) - sin(\frac{45}{2}))\cdot(2^{(\frac{3}{4})} - 8))}{(96ln(2))}|_{-6}^{-4}=0.05\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #141 : Triple Integration In Cartesian Coordinates

Example Question #282 : Multiple Integration

Example Question #281 : Triple Integrals

Example Question #284 : Multiple Integration

Example Question #285 : Multiple Integration

Example Question #286 : Multiple Integration

Example Question #287 : Multiple Integration

Example Question #288 : Multiple Integration

Example Question #289 : Multiple Integration

Example Question #290 : Multiple Integration

All Calculus 3 Resources

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