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Calculus 3 : Multiple Integration

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Example Questions

Example Question #411 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{13.5}\int_{3}^{7.5}\int_{2}^{5}(\frac{(5sin(z + 2))}{(8\cdot 3^{(\frac{y}{2})}x)})dxdydz\end{align*}$

Possible Answers:

-0.36

0.18

-0.09

0.03

Correct answer:

0.18

Explanation:

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

$\begin{align*}&\int_{9}^{13.5}\int_{3}^{7.5}\int_{2}^{5}(\frac{(5sin(z + 2))}{(8\cdot 3^{(\frac{y}{2})}x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{13.5}\int_{3}^{7.5}\int_{2}^{5}(\frac{(5sin(z + 2))}{(8\cdot 3^{(\frac{y}{2})}x)})dxdydz=\int_{9}^{13.5}\int_{3}^{7.5}(\frac{(5sin(z + 2)ln(x))}{(8\cdot 3^{(\frac{y}{2})})})dydz|_{2}^{5}\\&\int_{9}^{13.5}\int_{3}^{7.5}(\frac{(5sin(z + 2)ln(\frac{5}{2}))}{(8\cdot 3^{(\frac{y}{2})})})dydz=\int_{9}^{13.5}(-\frac{(5sin(z + 2)ln(\frac{5}{2}))}{(4\cdot 3^{(\frac{y}{2})}ln(3))})dz|_{3}^{7.5}\\&\int_{9}^{13.5}(\frac{(5sin(z + 2)ln(\frac{5}{2})\cdot (18\cdot 3^{(\frac{1}{2})} - 2\cdot 3^{(\frac{1}{4})}))}{(648ln(3))})dz=-\frac{(5cos(z + 2)ln(\frac{5}{2})\cdot (18\cdot 3^{(\frac{1}{2})} - 2\cdot 3^{(\frac{1}{4})}))}{(648ln(3))}|_{9}^{13.5}=0.18\end{align*}$

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Example Question #412 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-4.5}\int_{-3}^{1.5}\int_{3}^{5}(\frac{(11e^{(-2y)})}{(38\cdot 2^x)})dxdydz\end{align*}$

Possible Answers:

19.74

-19.74

-6.58

3.29

Correct answer:

19.74

Explanation:

$\begin{align*}&\int_{-7}^{-4.5}\int_{-3}^{1.5}\int_{3}^{5}(\frac{(11e^{(-2y)})}{(38\cdot 2^x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-4.5}\int_{-3}^{1.5}\int_{3}^{5}(\frac{(11e^{(-2y)})}{(38\cdot 2^x)})dxdydz=\int_{-7}^{-4.5}\int_{-3}^{1.5}(-\frac{(11e^{(-2y)})}{(38\cdot 2^xln(2))})dydz|_{3}^{5}\\&\int_{-7}^{-4.5}\int_{-3}^{1.5}(\frac{(33e^{(-2y)})}{(1216ln(2))})dydz=\int_{-7}^{-4.5}(-\frac{(33e^{(-2y)})}{(2432ln(2))})dz|_{-3}^{1.5}\\&\int_{-7}^{-4.5}(\frac{(33e^{(-3)}\cdot (e^{(9)} - 1))}{(2432ln(2))})dz=\frac{(33ze^{(-3)}\cdot (e^{(9)} - 1))}{(2432ln(2))}|_{-7}^{-4.5}=19.74\end{align*}$

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Example Question #413 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-5}^{-4}\int_{7}^{12}\int_{6}^{10.5}(\frac{(11sin(x + 1)e^{(-z)})}{(31y^{2})})dxdydz\end{align*}$

Possible Answers:

0.54

-1.07

-0.13

2.15

Correct answer:

0.54

Explanation:

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

$\begin{align*}&\int_{-5}^{-4}\int_{7}^{12}\int_{6}^{10.5}(\frac{(11sin(x + 1)e^{(-z)})}{(31y^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-5}^{-4}\int_{7}^{12}\int_{6}^{10.5}(\frac{(11sin(x + 1)e^{(-z)})}{(31y^{2})})dxdydz=\int_{-5}^{-4}\int_{7}^{12}(-\frac{(11cos(x + 1)e^{(-z)})}{(31y^{2})})dydz|_{6}^{10.5}\\&\int_{-5}^{-4}\int_{7}^{12}(\frac{(11e^{(-z)}\cdot (cos(7) - cos(\frac{23}{2})))}{(31y^{2})})dydz=\int_{-5}^{-4}(-\frac{(11e^{(-z)}\cdot (cos(7) - cos(\frac{23}{2})))}{(31y)})dz|_{7}^{12}\\&\int_{-5}^{-4}(\frac{(55e^{(-z)}\cdot (cos(7) - cos(\frac{23}{2})))}{2604})dz=-\frac{(55e^{(-z)}\cdot (cos(7) - cos(\frac{23}{2})))}{2604}|_{-5}^{-4}=0.54\end{align*}$

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Example Question #891 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{10}\int_{3.5}^{6.5}\int_{5}^{7}(\frac{(51e^{(-y)}e^{(x)})}{11})dxdydz\end{align*}$

Possible Answers:

-378.4

2271

378.4

Correct answer:

378.4

Explanation:

$\begin{align*}&\int_{7}^{10}\int_{3.5}^{6.5}\int_{5}^{7}(\frac{(51e^{(-y)}e^{(x)})}{11})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{10}\int_{3.5}^{6.5}\int_{5}^{7}(\frac{(51e^{(-y)}e^{(x)})}{11})dxdydz=\int_{7}^{10}\int_{3.5}^{6.5}(\frac{(51e^{(x - y)})}{11})dydz|_{5}^{7}\\&\int_{7}^{10}\int_{3.5}^{6.5}(\frac{(51e^{(5)}e^{(-y)}\cdot (e^{(2)} - 1))}{11})dydz=\int_{7}^{10}(-\frac{(51e^{(5)}e^{(-y)}\cdot (e^{(2)} - 1))}{11})dz|_{3.5}^{6.5}\\&\int_{7}^{10}(\frac{(51\cdot (e^{(\frac{1}{2})} - e^{(-\frac{3}{2})})\cdot (e^{(3)} - 1))}{11})dz=\frac{(51z\cdot (e^{(\frac{1}{2})} - e^{(-\frac{3}{2})})\cdot (e^{(3)} - 1))}{11}|_{7}^{10}=378.4\end{align*}$

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Example Question #281 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{12.5}\int_{7}^{9}\int_{8}^{11}(\frac{(5x^{2}cos(4y))}{z^{2}})dxdydz\end{align*}$

Possible Answers:

1.44

-17.24

-8.62

51.71

Correct answer:

-8.62

Explanation:

$\begin{align*}&\int_{10}^{12.5}\int_{7}^{9}\int_{8}^{11}(\frac{(5x^{2}cos(4y))}{z^{2}})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{12.5}\int_{7}^{9}\int_{8}^{11}(\frac{(5x^{2}cos(4y))}{z^{2}})dxdydz=\int_{10}^{12.5}\int_{7}^{9}(\frac{(5x^{3}cos(4y))}{(3z^{2})})dydz|_{8}^{11}\\&\int_{10}^{12.5}\int_{7}^{9}(\frac{(1365cos(4y))}{z^{2}})dydz=\int_{10}^{12.5}(\frac{(1365sin(4y))}{(4z^{2})})dz|_{7}^{9}\\&\int_{10}^{12.5}(-\frac{(1365\cdot (\frac{sin(28)}{4}-\frac{ sin(36)}{4}))}{z^{2}})dz=\frac{(\frac{(1365sin(28))}{4}-\frac{ (1365sin(36))}{4})}{z}|_{10}^{12.5}=-8.62\end{align*}$

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Example Question #281 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{5.5}\int_{7}^{11.5}\int_{-4.5}^{2.25}(\frac{(25e^{(-2x)})}{(48\cdot 2^{(\frac{z}{2})}y)})dxdydz\end{align*}$

Possible Answers:

-61.27

306.3

612.7

Correct answer:

306.3

Explanation:

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

$\begin{align*}&\int_{4}^{5.5}\int_{7}^{11.5}\int_{-4.5}^{2.25}(\frac{(25e^{(-2x)})}{(48\cdot 2^{(\frac{z}{2})}y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{5.5}\int_{7}^{11.5}\int_{-4.5}^{2.25}(\frac{(25e^{(-2x)})}{(48\cdot 2^{(\frac{z}{2})}y)})dxdydz=\int_{4}^{5.5}\int_{7}^{11.5}(-\frac{(25e^{(-2x)})}{(96\cdot 2^{(\frac{z}{2})}y)})dydz|_{-4.5}^{2.25}\\&\int_{4}^{5.5}\int_{7}^{11.5}(\frac{(25e^{(-\frac{9}{2})}\cdot (e^{(\frac{27}{2})} - 1))}{(96\cdot 2^{(\frac{z}{2})}y)})dydz=\int_{4}^{5.5}(\frac{(25e^{(-\frac{9}{2})}ln(y)\cdot (e^{(\frac{27}{2})} - 1))}{(96\cdot 2^{(\frac{z}{2})})})dz|_{7}^{11.5}\\&\int_{4}^{5.5}(\frac{(25e^{(-\frac{9}{2})}ln(\frac{23}{14})\cdot (e^{(\frac{27}{2})} - 1))}{(96\cdot 2^{(\frac{z}{2})})})dz=-\frac{(25e^{(-\frac{9}{2})}ln(\frac{23}{14})\cdot (e^{(\frac{27}{2})} - 1))}{(48\cdot 2^{(\frac{z}{2})}ln(2))}|_{4}^{5.5}=306.3\end{align*}$

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Example Question #282 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-3}\int_{10}^{11}\int_{-10}^{-7}(\frac{(18z)}{(5x)})dxdydz\end{align*}$

Possible Answers:

17.33

5.78

-86.67

-3.47

Correct answer:

17.33

Explanation:

$\begin{align*}&\int_{-6}^{-3}\int_{10}^{11}\int_{-10}^{-7}(\frac{(18z)}{(5x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-3}\int_{10}^{11}\int_{-10}^{-7}(\frac{(18z)}{(5x)})dxdydz=\int_{-6}^{-3}\int_{10}^{11}(\frac{(18zln(x))}{5})dydz|_{-10}^{-7}\\&\int_{-6}^{-3}\int_{10}^{11}(\frac{(18zln(\frac{7}{10}))}{5})dydz=\int_{-6}^{-3}(\frac{(18yzln(\frac{7}{10}))}{5})dz|_{10}^{11}\\&\int_{-6}^{-3}(\frac{(18zln(\frac{7}{10}))}{5})dz=\frac{(9z^{2}ln(\frac{7}{10}))}{5}|_{-6}^{-3}=17.33\end{align*}$

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Example Question #283 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{11.5}\int_{-3.5}^{1.5}\int_{9}^{13.5}(25sin(3x)sin(4z)e^{(2y)})dxdydz\end{align*}$

Possible Answers:

0.64

-19.17

-3.19

19.17

Correct answer:

-3.19

Explanation:

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

$\begin{align*}&\int_{10}^{11.5}\int_{-3.5}^{1.5}\int_{9}^{13.5}(25sin(3x)sin(4z)e^{(2y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{11.5}\int_{-3.5}^{1.5}\int_{9}^{13.5}(25sin(3x)sin(4z)e^{(2y)})dxdydz=\int_{10}^{11.5}\int_{-3.5}^{1.5}(-\frac{(25cos(3x)sin(4z)e^{(2y)})}{3})dydz|_{9}^{13.5}\\&\int_{10}^{11.5}\int_{-3.5}^{1.5}(25sin(4z)e^{(2y)}\cdot (\frac{cos(27)}{3}-\frac{ cos(\frac{81}{2})}{3}))dydz=\int_{10}^{11.5}(\frac{(25sin(4z)e^{(2y)}\cdot (\frac{cos(27)}{3}-\frac{ cos(\frac{81}{2})}{3}))}{2})dz|_{-3.5}^{1.5}\\&\int_{10}^{11.5}(\frac{(25sin(4z)e^{(-7)}\cdot (cos(27) - cos(\frac{81}{2}))\cdot (e^{(10)} - 1))}{6})dz=-\frac{(25cos(4z)e^{(-7)}\cdot (cos(27) - cos(\frac{81}{2}))\cdot (e^{(10)} - 1))}{24}|_{10}^{11.5}=-3.19\end{align*}$

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Example Question #282 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-6}\int_{10}^{13}\int_{6}^{11}(\frac{(59x^{2}sin(4y))}{(7z^{2})})dxdydz\end{align*}$

Possible Answers:

1.57

37.59

-18.79

-9.4

Correct answer:

-9.4

Explanation:

$\begin{align*}&\int_{-7}^{-6}\int_{10}^{13}\int_{6}^{11}(\frac{(59x^{2}sin(4y))}{(7z^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-6}\int_{10}^{13}\int_{6}^{11}(\frac{(59x^{2}sin(4y))}{(7z^{2})})dxdydz=\int_{-7}^{-6}\int_{10}^{13}(\frac{(59x^{3}sin(4y))}{(21z^{2})})dydz|_{6}^{11}\\&\int_{-7}^{-6}\int_{10}^{13}(\frac{(65785sin(4y))}{(21z^{2})})dydz=\int_{-7}^{-6}(-\frac{(65785cos(4y))}{(84z^{2})})dz|_{10}^{13}\\&\int_{-7}^{-6}(\frac{(65785\cdot (\frac{cos(40)}{4}-\frac{ cos(52)}{4}))}{(21z^{2})})dz=-\frac{(\frac{(65785cos(40))}{84}-\frac{ (65785cos(52))}{84})}{z}|_{-7}^{-6}=-9.4\end{align*}$

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Example Question #891 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{-8}^{-4.5}\int_{-4.5}^{-1}\int_{-4}^{1}(\frac{(e^{(2x)}e^{(-y)})}{(2z)})dxdydz\end{align*}$

Possible Answers:

185.6

23.2

-463.9

-92.78

Correct answer:

-92.78

Explanation:

$\begin{align*}&\int_{-8}^{-4.5}\int_{-4.5}^{-1}\int_{-4}^{1}(\frac{(e^{(2x)}e^{(-y)})}{(2z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-8}^{-4.5}\int_{-4.5}^{-1}\int_{-4}^{1}(\frac{(e^{(2x)}e^{(-y)})}{(2z)})dxdydz=\int_{-8}^{-4.5}\int_{-4.5}^{-1}(\frac{e^{(2x - y)}}{(4z)})dydz|_{-4}^{1}\\&\int_{-8}^{-4.5}\int_{-4.5}^{-1}(\frac{(e^{(-8)}e^{(-y)}\cdot (e^{(10)} - 1))}{(4z)})dydz=\int_{-8}^{-4.5}(-\frac{(e^{(- y - 8)}\cdot (e^{(10)} - 1))}{(4z)})dz|_{-4.5}^{-1}\\&\int_{-8}^{-4.5}(\frac{(e^{(-7)}\cdot (e^{(\frac{7}{2})} - 1)\cdot (e^{(10)} - 1))}{(4z)})dz=-\frac{(e^{(-7)}ln(z)\cdot (e^{(\frac{7}{2})} + e^{(10)} - e^{(\frac{27}{2})} - 1))}{4}|_{-8}^{-4.5}=-92.78\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #411 : Triple Integrals

Example Question #412 : Triple Integrals

Example Question #413 : Triple Integrals

Example Question #891 : Calculus 3

Example Question #281 : Triple Integration In Cartesian Coordinates

Example Question #281 : Triple Integration In Cartesian Coordinates

Example Question #282 : Triple Integration In Cartesian Coordinates

Example Question #283 : Triple Integration In Cartesian Coordinates

Example Question #282 : Triple Integration In Cartesian Coordinates

Example Question #891 : Calculus 3

All Calculus 3 Resources

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