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Calculus 3 : Multiple Integration

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Example Questions

Example Question #321 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-8}\int_{8}^{11.5}\int_{9}^{10}(\frac{(37z)}{(6x)})dxdydz\end{align*}$

Possible Answers:

-38.66

-19.33

9.66

57.99

Correct answer:

-19.33

Explanation:

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

$\begin{align*}&\int_{-9}^{-8}\int_{8}^{11.5}\int_{9}^{10}(\frac{(37z)}{(6x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-8}\int_{8}^{11.5}\int_{9}^{10}(\frac{(37z)}{(6x)})dxdydz=\int_{-9}^{-8}\int_{8}^{11.5}(\frac{(37zln(x))}{6})dydz|_{9}^{10}\\&\int_{-9}^{-8}\int_{8}^{11.5}(\frac{(37zln(\frac{10}{9}))}{6})dydz=\int_{-9}^{-8}(\frac{(37yzln(\frac{10}{9}))}{6})dz|_{8}^{11.5}\\&\int_{-9}^{-8}(\frac{(259zln(\frac{10}{9}))}{12})dz=\frac{(259z^{2}ln(\frac{10}{9}))}{24}|_{-9}^{-8}=-19.33\end{align*}$

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Example Question #322 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{5}^{8}\int_{-3.5}^{1}\int_{-8}^{-5}(\frac{e^{(2z)}}{(7\cdot2^yx^{2})})dxdydz\end{align*}$

Possible Answers:

$-1.482\cdot10^{6}$

246900

740800

Correct answer:

740800

Explanation:

$\begin{align*}&\int_{5}^{8}\int_{-3.5}^{1}\int_{-8}^{-5}(\frac{e^{(2z)}}{(7\cdot2^yx^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{5}^{8}\int_{-3.5}^{1}\int_{-8}^{-5}(\frac{e^{(2z)}}{(7\cdot2^yx^{2})})dxdydz=\int_{5}^{8}\int_{-3.5}^{1}(-\frac{e^{(2z)}}{(7\cdot2^yx)})dydz|_{-8}^{-5}\\&\int_{5}^{8}\int_{-3.5}^{1}(\frac{(3e^{(2z)})}{(280\cdot2^y)})dydz=\int_{5}^{8}(-\frac{(3e^{(2z)})}{(280\cdot2^yln(2))})dz|_{-3.5}^{1}\\&\int_{5}^{8}(\frac{(3e^{(2z)}\cdot(8\cdot2^{(\frac{1}{2})} -\frac{ 1}{2}))}{(280ln(2))})dz=\frac{(3e^{(2z)}\cdot(8\cdot2^{(\frac{1}{2})} -\frac{ 1}{2}))}{(560ln(2))}|_{5}^{8}=740800\end{align*}$

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Example Question #321 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{10.5}\int_{-8}^{-4.5}\int_{10}^{14}(\frac{(3ycos(4x)sin(z + 2))}{5})dxdydz\end{align*}$

Possible Answers:

-2.65

15.87

-7.93

3.97

Correct answer:

-7.93

Explanation:

$\begin{align*}&\int_{7}^{10.5}\int_{-8}^{-4.5}\int_{10}^{14}(\frac{(3ycos(4x)sin(z + 2))}{5})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{10.5}\int_{-8}^{-4.5}\int_{10}^{14}(\frac{(3ycos(4x)sin(z + 2))}{5})dxdydz=\int_{7}^{10.5}\int_{-8}^{-4.5}(\frac{(3ysin(4x)sin(z + 2))}{20})dydz|_{10}^{14}\\&\int_{7}^{10.5}\int_{-8}^{-4.5}(-\frac{(3ysin(z + 2)\cdot(sin(40) - sin(56)))}{20})dydz=\int_{7}^{10.5}(-\frac{(3y^{2}sin(z + 2)\cdot(sin(40) - sin(56)))}{40})dz|_{-8}^{-4.5}\\&\int_{7}^{10.5}(\frac{(105sin(z + 2)\cdot(sin(40) - sin(56)))}{32})dz=-cos(z + 2)\cdot(\frac{(105sin(40))}{32}-\frac{ (105sin(56))}{32})|_{7}^{10.5}=-7.93\end{align*}$

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Example Question #191 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-7.5}\int_{4.5}^{8}\int_{8}^{12.5}(\frac{(6\cdot3^ycos(z + 2)sin(4x))}{49})dxdydz\end{align*}$

Possible Answers:

7.97

127.5

-31.87

-159.4

Correct answer:

-31.87

Explanation:

$\begin{align*}&\int_{-9}^{-7.5}\int_{4.5}^{8}\int_{8}^{12.5}(\frac{(6\cdot3^ycos(z + 2)sin(4x))}{49})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-7.5}\int_{4.5}^{8}\int_{8}^{12.5}(\frac{(6\cdot3^ycos(z + 2)sin(4x))}{49})dxdydz=\int_{-9}^{-7.5}\int_{4.5}^{8}(-\frac{(3\cdot3^ycos(4x)cos(z + 2))}{98})dydz|_{8}^{12.5}\\&\int_{-9}^{-7.5}\int_{4.5}^{8}(\frac{(6\cdot3^ycos(z + 2)\cdot(\frac{cos(32)}{4}-\frac{ cos(50)}{4}))}{49})dydz=\int_{-9}^{-7.5}(\frac{(3\cdot3^y\cdot(cos(z + 2)cos(32) - cos(z + 2)cos(50)))}{(98ln(3))})dz|_{4.5}^{8}\\&\int_{-9}^{-7.5}(-\frac{(243cos(z + 2)\cdot(cos(32) - cos(50))\cdot(3^{(\frac{1}{2})} - 81))}{(98ln(3))})dz=-\frac{(243sin(z + 2)\cdot(cos(32) - cos(50))\cdot(3^{(\frac{1}{2})} - 81))}{(98ln(3))}|_{-9}^{-7.5}=-31.87\end{align*}$

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Example Question #192 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{11}\int_{-8}^{-5.5}\int_{-5}^{-4}(\frac{(13z^{2}e^{(-2x)})}{(7y)})dxdydz\end{align*}$

Possible Answers:

$2.182\cdot10^{6}$

$-2.182\cdot10^{6}$

436500

$-1.309\cdot10^{7}$

Correct answer:

$-2.182\cdot10^{6}$

Explanation:

$\begin{align*}&\int_{7}^{11}\int_{-8}^{-5.5}\int_{-5}^{-4}(\frac{(13z^{2}e^{(-2x)})}{(7y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{11}\int_{-8}^{-5.5}\int_{-5}^{-4}(\frac{(13z^{2}e^{(-2x)})}{(7y)})dxdydz=\int_{7}^{11}\int_{-8}^{-5.5}(-\frac{(13z^{2}e^{(-2x)})}{(14y)})dydz|_{-5}^{-4}\\&\int_{7}^{11}\int_{-8}^{-5.5}(\frac{(13z^{2}e^{(8)}\cdot(e^{(2)} - 1))}{(14y)})dydz=\int_{7}^{11}(\frac{(13z^{2}e^{(8)}ln(y)\cdot(e^{(2)} - 1))}{14})dz|_{-8}^{-5.5}\\&\int_{7}^{11}(\frac{(13z^{2}e^{(8)}ln(\frac{11}{16})\cdot(e^{(2)} - 1))}{14})dz=\frac{(13z^{3}e^{(8)}ln(\frac{11}{16})\cdot(e^{(2)} - 1))}{42}|_{7}^{11}=-2.182\cdot10^{6}\end{align*}$

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Example Question #193 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{9}\int_{-5}^{-2.5}\int_{8}^{11.5}(\frac{(9xe^{(y)})}{(11z)})dxdydz\end{align*}$

Possible Answers:

-0.11

0.53

1.06

-0.53

Correct answer:

0.53

Explanation:

$\begin{align*}&\int_{7}^{9}\int_{-5}^{-2.5}\int_{8}^{11.5}(\frac{(9xe^{(y)})}{(11z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{9}\int_{-5}^{-2.5}\int_{8}^{11.5}(\frac{(9xe^{(y)})}{(11z)})dxdydz=\int_{7}^{9}\int_{-5}^{-2.5}(\frac{(9x^{2}e^{(y)})}{(22z)})dydz|_{8}^{11.5}\\&\int_{7}^{9}\int_{-5}^{-2.5}(\frac{(2457e^{(y)})}{(88z)})dydz=\int_{7}^{9}(\frac{(2457e^{(y)})}{(88z)})dz|_{-5}^{-2.5}\\&\int_{7}^{9}(\frac{(2457e^{(-5)}\cdot(e^{(\frac{5}{2})} - 1))}{(88z)})dz=\frac{(2457e^{(-5)}ln(z)\cdot(e^{(\frac{5}{2})} - 1))}{88}|_{7}^{9}=0.53\end{align*}$

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Example Question #194 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{5.5}\int_{-10}^{-5.5}\int_{6}^{9.5}(27cos(3y)sin(4x)e^{(2z)})dxdydz\end{align*}$

Possible Answers:

46933

9387

Correct answer:

9387

Explanation:

$\begin{align*}&\int_{4}^{5.5}\int_{-10}^{-5.5}\int_{6}^{9.5}(27cos(3y)sin(4x)e^{(2z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{5.5}\int_{-10}^{-5.5}\int_{6}^{9.5}(27cos(3y)sin(4x)e^{(2z)})dxdydz=\int_{4}^{5.5}\int_{-10}^{-5.5}(-\frac{(27cos(4x)cos(3y)e^{(2z)})}{4})dydz|_{6}^{9.5}\\&\int_{4}^{5.5}\int_{-10}^{-5.5}(\frac{(27cos(3y)e^{(2z)}\cdot(cos(24) - cos(38)))}{4})dydz=\int_{4}^{5.5}(\frac{(9sin(3y)e^{(2z)}\cdot(cos(24) - cos(38)))}{4})dz|_{-10}^{-5.5}\\&\int_{4}^{5.5}(\frac{(9e^{(2z)}\cdot(cos(24) - cos(38))\cdot(sin(30) - sin(\frac{33}{2})))}{4})dz=\frac{(9e^{(2z)}\cdot(cos(24) - cos(38))\cdot(sin(30) - sin(\frac{33}{2})))}{8}|_{4}^{5.5}=9387\end{align*}$

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Example Question #195 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{8}^{11}\int_{9}^{13}\int_{-10}^{-6.5}(\frac{(3y^{2})}{2})dxdydz\end{align*}$

Possible Answers:

1927

7707

Correct answer:

7707

Explanation:

$\begin{align*}&\int_{8}^{11}\int_{9}^{13}\int_{-10}^{-6.5}(\frac{(3y^{2})}{2})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{8}^{11}\int_{9}^{13}\int_{-10}^{-6.5}(\frac{(3y^{2})}{2})dxdydz=\int_{8}^{11}\int_{9}^{13}(\frac{(3xy^{2})}{2})dydz|_{-10}^{-6.5}\\&\int_{8}^{11}\int_{9}^{13}(\frac{(21y^{2})}{4})dydz=\int_{8}^{11}(\frac{(7y^{3})}{4})dz|_{9}^{13}\\&\int_{8}^{11}(2569)dz=2569z|_{8}^{11}=7707\end{align*}$

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Example Question #196 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{10}\int_{7}^{8}\int_{-5}^{-3.5}(\frac{(17y^{2})}{(21\cdot2^xz)})dxdydz\end{align*}$

Possible Answers:

695.2

-115.9

115.9

Correct answer:

695.2

Explanation:

$\begin{align*}&\int_{6}^{10}\int_{7}^{8}\int_{-5}^{-3.5}(\frac{(17y^{2})}{(21\cdot2^xz)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{10}\int_{7}^{8}\int_{-5}^{-3.5}(\frac{(17y^{2})}{(21\cdot2^xz)})dxdydz=\int_{6}^{10}\int_{7}^{8}(-\frac{(17y^{2})}{(21\cdot2^xzln(2))})dydz|_{-5}^{-3.5}\\&\int_{6}^{10}\int_{7}^{8}(-\frac{(136y^{2}\cdot(2^{(\frac{1}{2})} - 4))}{(21zln(2))})dydz=\int_{6}^{10}(-\frac{(y^{3}\cdot(136\cdot2^{(\frac{1}{2})} - 544))}{(63zln(2))})dz|_{7}^{8}\\&\int_{6}^{10}(-\frac{(22984\cdot(2^{(\frac{1}{2})} - 4))}{(63zln(2))})dz=-\frac{(ln(z)\cdot(22984\cdot2^{(\frac{1}{2})} - 91936))}{(63ln(2))}|_{6}^{10}=695.2\end{align*}$

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Example Question #197 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{10.5}\int_{8}^{9}\int_{-4.5}^{-2.5}(\frac{1}{(9\cdot3^xz^{3})})dxdydz\end{align*}$

Possible Answers:

-0.02

0.12

0.47

-0.24

Correct answer:

0.12

Explanation:

$\begin{align*}&\int_{6}^{10.5}\int_{8}^{9}\int_{-4.5}^{-2.5}(\frac{1}{(9\cdot3^xz^{3})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{10.5}\int_{8}^{9}\int_{-4.5}^{-2.5}(\frac{1}{(9\cdot3^xz^{3})})dxdydz=\int_{6}^{10.5}\int_{8}^{9}(-\frac{1}{(9\cdot3^xz^{3}ln(3))})dydz|_{-4.5}^{-2.5}\\&\int_{6}^{10.5}\int_{8}^{9}(\frac{(8\cdot3^{(\frac{1}{2})})}{(z^{3}ln(3))})dydz=\int_{6}^{10.5}(\frac{(8\cdot3^{(\frac{1}{2})}y)}{(z^{3}ln(3))})dz|_{8}^{9}\\&\int_{6}^{10.5}(\frac{(8\cdot3^{(\frac{1}{2})})}{(z^{3}ln(3))})dz=-\frac{(4\cdot3^{(\frac{1}{2})})}{(z^{2}ln(3))}|_{6}^{10.5}=0.12\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #321 : Multiple Integration

Example Question #322 : Multiple Integration

Example Question #321 : Triple Integrals

Example Question #191 : Triple Integration In Cartesian Coordinates

Example Question #192 : Triple Integration In Cartesian Coordinates

Example Question #193 : Triple Integration In Cartesian Coordinates

Example Question #194 : Triple Integration In Cartesian Coordinates

Example Question #195 : Triple Integration In Cartesian Coordinates

Example Question #196 : Triple Integration In Cartesian Coordinates

Example Question #197 : Triple Integration In Cartesian Coordinates

All Calculus 3 Resources

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