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Calculus 3 : Multiple Integration

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Example Questions

Example Question #291 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{10.5}\int_{-7}^{-3.5}\int_{-4}^{-1}(\frac{(23e^{(-2x)})}{(9yz^{2})})dxdydz\end{align*}$

Possible Answers:

20.9

-250.8

41.8

-41.8

Correct answer:

-41.8

Explanation:

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

$\begin{align*}&\int_{9}^{10.5}\int_{-7}^{-3.5}\int_{-4}^{-1}(\frac{(23e^{(-2x)})}{(9yz^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{10.5}\int_{-7}^{-3.5}\int_{-4}^{-1}(\frac{(23e^{(-2x)})}{(9yz^{2})})dxdydz=\int_{9}^{10.5}\int_{-7}^{-3.5}(-\frac{(23e^{(-2x)})}{(18yz^{2})})dydz|_{-4}^{-1}\\&\int_{9}^{10.5}\int_{-7}^{-3.5}(\frac{(23e^{(2)}\cdot (e^{(6)} - 1))}{(18yz^{2})})dydz=\int_{9}^{10.5}(\frac{(23e^{(2)}ln(y)\cdot (e^{(6)} - 1))}{(18z^{2})})dz|_{-7}^{-3.5}\\&\int_{9}^{10.5}(-\frac{(23e^{(2)}ln(2)\cdot (e^{(6)} - 1))}{(18z^{2})})dz=\frac{(23e^{(2)}ln(2)\cdot (e^{(6)} - 1))}{(18z)}|_{9}^{10.5}=-41.8\end{align*}$

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Example Question #431 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{8}^{11.5}\int_{-3.5}^{-1.5}\int_{3}^{5}(\frac{(4\cdot 2^{(\frac{x}{4})}\cdot 3^ycos(3z))}{3})dxdydz\end{align*}$

Possible Answers:

-1.34

-0.07

0.27

0.54

Correct answer:

0.27

Explanation:

$\begin{align*}&\int_{8}^{11.5}\int_{-3.5}^{-1.5}\int_{3}^{5}(\frac{(4\cdot 2^{(\frac{x}{4})}\cdot 3^ycos(3z))}{3})dxdydz\\&\text{The approach is to take it step by step:}\\&\int_{8}^{11.5}\int_{-3.5}^{-1.5}\int_{3}^{5}(\frac{(4\cdot 2^{(\frac{x}{4})}\cdot 3^ycos(3z))}{3})dxdydz=\int_{8}^{11.5}\int_{-3.5}^{-1.5}(\frac{(16\cdot 2^{(\frac{x}{4})}\cdot 3^{(y - 1)}cos(3z))}{ln(2)})dydz|_{3}^{5}\\&\int_{8}^{11.5}\int_{-3.5}^{-1.5}(\frac{(4\cdot 3^ycos(3z)\cdot (8\cdot 2^{(\frac{1}{4})} - 4\cdot 2^{(\frac{3}{4})}))}{(3ln(2))})dydz=\int_{8}^{11.5}(\frac{(16\cdot 3^y\cdot (2\cdot 2^{(\frac{1}{4})}cos(3z) - 8^{(\frac{1}{4})}cos(3z)))}{(3ln(2)ln(3))})dz|_{-3.5}^{-1.5}\\&\int_{8}^{11.5}(\frac{(256\cdot 2^{(\frac{1}{4})}\cdot 3^{(\frac{1}{2})}cos(3z) - 128\cdot 2^{(\frac{3}{4})}\cdot 3^{(\frac{1}{2})}cos(3z))}{(243ln(2)ln(3))})dz=\frac{(256\cdot 2^{(\frac{1}{4})}\cdot 3^{(\frac{1}{2})}sin(3z) - 128\cdot 3^{(\frac{1}{2})}\cdot 8^{(\frac{1}{4})}sin(3z))}{(729ln(2)ln(3))}|_{8}^{11.5}=0.27\end{align*}$

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Example Question #432 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{4.5}^{8}\int_{-7}^{-5.5}\int_{-6}^{-4.5}(\frac{(18\cdot 3^zsin(x + 2)sin(3y))}{11})dxdydz\end{align*}$

Possible Answers:

72.73

-72.73

12.12

-12.12

Correct answer:

72.73

Explanation:

$\begin{align*}&\int_{4.5}^{8}\int_{-7}^{-5.5}\int_{-6}^{-4.5}(\frac{(18\cdot 3^zsin(x + 2)sin(3y))}{11})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4.5}^{8}\int_{-7}^{-5.5}\int_{-6}^{-4.5}(\frac{(18\cdot 3^zsin(x + 2)sin(3y))}{11})dxdydz=\int_{4.5}^{8}\int_{-7}^{-5.5}(-\frac{(18\cdot 3^zcos(x + 2)sin(3y))}{11})dydz|_{-6}^{-4.5}\\&\int_{4.5}^{8}\int_{-7}^{-5.5}(\frac{(18\cdot 3^zsin(3y)\cdot (cos(4) - cos(\frac{5}{2})))}{11})dydz=\int_{4.5}^{8}(-\frac{(6\cdot 3^zcos(3y)\cdot (cos(4) - cos(\frac{5}{2})))}{11})dz|_{-7}^{-5.5}\\&\int_{4.5}^{8}(\frac{(6\cdot 3^z\cdot (cos(4) - cos(\frac{5}{2}))\cdot (cos(21) - cos(\frac{33}{2})))}{11})dz=\frac{(6\cdot 3^z\cdot (cos(4)cos(21) - cos(\frac{5}{2})cos(21) - cos(4)cos(\frac{33}{2}) + cos(\frac{5}{2})cos(\frac{33}{2})))}{(11ln(3))}|_{4.5}^{8}=72.73\end{align*}$

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Example Question #431 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{10}\int_{5}^{9.5}\int_{-10}^{-8}(\frac{(6z^{2}sin(x + 1)e^{(-y)})}{49})dxdydz\end{align*}$

Possible Answers:

0.18

-1.77

2.13

-0.35

Correct answer:

-0.35

Explanation:

$\begin{align*}&\int_{6}^{10}\int_{5}^{9.5}\int_{-10}^{-8}(\frac{(6z^{2}sin(x + 1)e^{(-y)})}{49})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{10}\int_{5}^{9.5}\int_{-10}^{-8}(\frac{(6z^{2}sin(x + 1)e^{(-y)})}{49})dxdydz=\int_{6}^{10}\int_{5}^{9.5}(-\frac{(6z^{2}cos(x + 1)e^{(-y)})}{49})dydz|_{-10}^{-8}\\&\int_{6}^{10}\int_{5}^{9.5}(-\frac{(6z^{2}e^{(-y)}\cdot (cos(7) - cos(9)))}{49})dydz=\int_{6}^{10}(\frac{(6z^{2}e^{(-y)}\cdot (cos(7) - cos(9)))}{49})dz|_{5}^{9.5}\\&\int_{6}^{10}(-\frac{(6z^{2}e^{(-\frac{19}{2})}\cdot (cos(7) - cos(9))\cdot (e^{(\frac{9}{2})} - 1))}{49})dz=-z^{3}\cdot (\frac{(2cos(7)e^{(-5)})}{49}-\frac{ (2cos(9)e^{(-5)})}{49}-\frac{ (2cos(7)e^{(-\frac{19}{2})})}{49}+\frac{ (2cos(9)e^{(-\frac{19}{2})})}{49})|_{6}^{10}=-0.35\end{align*}$

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Example Question #432 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{7}\int_{7}^{9.5}\int_{-6}^{-2.5}(\frac{(3y^{2}cos(x + 1)e^{(-2z)})}{53})dxdydz\end{align*}$

Possible Answers:

-0.12

-0.09

0.09

-0.02

Correct answer:

-0.02

Explanation:

$\begin{align*}&\int_{3}^{7}\int_{7}^{9.5}\int_{-6}^{-2.5}(\frac{(3y^{2}cos(x + 1)e^{(-2z)})}{53})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{7}\int_{7}^{9.5}\int_{-6}^{-2.5}(\frac{(3y^{2}cos(x + 1)e^{(-2z)})}{53})dxdydz=\int_{3}^{7}\int_{7}^{9.5}(\frac{(3y^{2}sin(x + 1)e^{(-2z)})}{53})dydz|_{-6}^{-2.5}\\&\int_{3}^{7}\int_{7}^{9.5}(-\frac{(3y^{2}e^{(-2z)}\cdot (sin(\frac{3}{2}) - sin(5)))}{53})dydz=\int_{3}^{7}(-y^{3}\cdot (\frac{(e^{(-2z)}sin(\frac{3}{2}))}{53}-\frac{ (e^{(-2z)}sin(5))}{53}))dz|_{7}^{9.5}\\&\int_{3}^{7}(-\frac{(4115e^{(-2z)}\cdot (sin(\frac{3}{2}) - sin(5)))}{424})dz=\frac{(4115e^{(-2z)}\cdot (sin(\frac{3}{2}) - sin(5)))}{848}|_{3}^{7}=-0.02\end{align*}$

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Example Question #431 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-6.5}\int_{6}^{11}\int_{-3}^{-0.5}(\frac{(57e^{(2x)})}{(5y^{2})})dxdydz\end{align*}$

Possible Answers:

-0.1

0.1

-1.18

0.39

Correct answer:

0.39

Explanation:

$\begin{align*}&\int_{-9}^{-6.5}\int_{6}^{11}\int_{-3}^{-0.5}(\frac{(57e^{(2x)})}{(5y^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-6.5}\int_{6}^{11}\int_{-3}^{-0.5}(\frac{(57e^{(2x)})}{(5y^{2})})dxdydz=\int_{-9}^{-6.5}\int_{6}^{11}(\frac{(57e^{(2x)})}{(10y^{2})})dydz|_{-3}^{-0.5}\\&\int_{-9}^{-6.5}\int_{6}^{11}(\frac{(57e^{(-6)}\cdot (e^{(5)} - 1))}{(10y^{2})})dydz=\int_{-9}^{-6.5}(-\frac{(57e^{(-6)}\cdot (e^{(5)} - 1))}{(10y)})dz|_{6}^{11}\\&\int_{-9}^{-6.5}(\frac{(19e^{(-6)}\cdot (e^{(5)} - 1))}{44})dz=\frac{(19ze^{(-6)}\cdot (e^{(5)} - 1))}{44}|_{-9}^{-6.5}=0.39\end{align*}$

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Example Question #434 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{11.5}\int_{-6}^{-4}\int_{-8}^{-3.5}(\frac{(5sin(4y)sin(z + 2))}{x})dxdydz\end{align*}$

Possible Answers:

0.07

-1.42

1.07

-0.36

Correct answer:

-0.36

Explanation:

$\begin{align*}&\int_{10}^{11.5}\int_{-6}^{-4}\int_{-8}^{-3.5}(\frac{(5sin(4y)sin(z + 2))}{x})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{11.5}\int_{-6}^{-4}\int_{-8}^{-3.5}(\frac{(5sin(4y)sin(z + 2))}{x})dxdydz=\int_{10}^{11.5}\int_{-6}^{-4}(5sin(4y)sin(z + 2)ln(x))dydz|_{-8}^{-3.5}\\&\int_{10}^{11.5}\int_{-6}^{-4}(5sin(4y)sin(z + 2)ln(\frac{7}{16}))dydz=\int_{10}^{11.5}(-\frac{(5cos(4y)sin(z + 2)ln(\frac{7}{16}))}{4})dz|_{-6}^{-4}\\&\int_{10}^{11.5}(-5sin(z + 2)ln(\frac{7}{16})\cdot (\frac{cos(16)}{4}-\frac{ cos(24)}{4}))dz=5cos(z + 2)ln(\frac{7}{16})\cdot (\frac{cos(16)}{4}-\frac{ cos(24)}{4})|_{10}^{11.5}=-0.36\end{align*}$

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Example Question #431 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{8}^{10}\int_{-9}^{-8}\int_{7}^{10.5}(\frac{(23y^{2}cos(4z))}{(5x)})dxdydz\end{align*}$

Possible Answers:

-1.09

-26.13

6.53

19.6

Correct answer:

6.53

Explanation:

$\begin{align*}&\int_{8}^{10}\int_{-9}^{-8}\int_{7}^{10.5}(\frac{(23y^{2}cos(4z))}{(5x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{8}^{10}\int_{-9}^{-8}\int_{7}^{10.5}(\frac{(23y^{2}cos(4z))}{(5x)})dxdydz=\int_{8}^{10}\int_{-9}^{-8}(\frac{(23y^{2}cos(4z)ln(x))}{5})dydz|_{7}^{10.5}\\&\int_{8}^{10}\int_{-9}^{-8}(\frac{(23y^{2}cos(4z)ln(\frac{3}{2}))}{5})dydz=\int_{8}^{10}(\frac{(23y^{3}cos(4z)ln(\frac{3}{2}))}{15})dz|_{-9}^{-8}\\&\int_{8}^{10}(\frac{(4991cos(4z)ln(\frac{3}{2}))}{15})dz=\frac{(4991sin(4z)ln(\frac{3}{2}))}{60}|_{8}^{10}=6.53\end{align*}$

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Example Question #435 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-10}^{-7.5}\int_{7}^{9}\int_{-3.5}^{1}(\frac{(sin(y + 1)e^{(2x)})}{(92z)})dxdydz\end{align*}$

Possible Answers:

-0.05

0.91

-0.09

-0.01

Correct answer:

-0.01

Explanation:

$\begin{align*}&\int_{-10}^{-7.5}\int_{7}^{9}\int_{-3.5}^{1}(\frac{(sin(y + 1)e^{(2x)})}{(92z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-10}^{-7.5}\int_{7}^{9}\int_{-3.5}^{1}(\frac{(sin(y + 1)e^{(2x)})}{(92z)})dxdydz=\int_{-10}^{-7.5}\int_{7}^{9}(\frac{(sin(y + 1)e^{(2x)})}{(184z)})dydz|_{-3.5}^{1}\\&\int_{-10}^{-7.5}\int_{7}^{9}(\frac{(sin(y + 1)e^{(-7)}\cdot (e^{(9)} - 1))}{(184z)})dydz=\int_{-10}^{-7.5}(-\frac{(cos(y + 1)e^{(-7)}\cdot (e^{(9)} - 1))}{(184z)})dz|_{7}^{9}\\&\int_{-10}^{-7.5}(\frac{(e^{(-7)}\cdot (cos(8) - cos(10))\cdot (e^{(9)} - 1))}{(184z)})dz=-\frac{(e^{(-7)}ln(z)\cdot (cos(8) - cos(10) - cos(8)e^{(9)} + cos(10)e^{(9)}))}{184}|_{-10}^{-7.5}=-0.01\end{align*}$

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Example Question #436 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3.5}^{-0.5}\int_{-10}^{-5}\int_{-3.5}^{1.5}(19e^{(-x)}e^{(z)})dxdydz\end{align*}$

Possible Answers:

1801

900.5

-450.2

Correct answer:

1801

Explanation:

$\begin{align*}&\int_{-3.5}^{-0.5}\int_{-10}^{-5}\int_{-3.5}^{1.5}(19e^{(-x)}e^{(z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3.5}^{-0.5}\int_{-10}^{-5}\int_{-3.5}^{1.5}(19e^{(-x)}e^{(z)})dxdydz=\int_{-3.5}^{-0.5}\int_{-10}^{-5}(-19e^{(z - x)})dydz|_{-3.5}^{1.5}\\&\int_{-3.5}^{-0.5}\int_{-10}^{-5}(19e^{(-\frac{3}{2})}e^{(z)}\cdot (e^{(5)} - 1))dydz=\int_{-3.5}^{-0.5}(19ye^{(-\frac{3}{2})}e^{(z)}\cdot (e^{(5)} - 1))dz|_{-10}^{-5}\\&\int_{-3.5}^{-0.5}(95e^{(-\frac{3}{2})}e^{(z)}\cdot (e^{(5)} - 1))dz=95e^{(-\frac{3}{2})}e^{(z)}\cdot (e^{(5)} - 1)|_{-3.5}^{-0.5}=1801\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #291 : Triple Integration In Cartesian Coordinates

Example Question #431 : Multiple Integration

Example Question #432 : Multiple Integration

Example Question #431 : Multiple Integration

Example Question #432 : Multiple Integration

Example Question #431 : Triple Integrals

Example Question #434 : Multiple Integration

Example Question #431 : Multiple Integration

Example Question #435 : Multiple Integration

Example Question #436 : Multiple Integration

All Calculus 3 Resources

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