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Calculus 3 : Multiple Integration

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Example Questions

Example Question #231 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-10}^{-7}\int_{7}^{9}\int_{9}^{11}(\frac{(4sin(3z)e^{(y)})}{(17x^{2})})dxdydz\end{align*}$

Possible Answers:

-23.38

1.3

7.79

38.97

Correct answer:

7.79

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-10}^{-7}\int_{7}^{9}\int_{9}^{11}(\frac{(4sin(3z)e^{(y)})}{(17x^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-10}^{-7}\int_{7}^{9}\int_{9}^{11}(\frac{(4sin(3z)e^{(y)})}{(17x^{2})})dxdydz=\int_{-10}^{-7}\int_{7}^{9}(-\frac{(4sin(3z)e^{(y)})}{(17x)})dydz|_{9}^{11}\\&\int_{-10}^{-7}\int_{7}^{9}(\frac{(8sin(3z)e^{(y)})}{1683})dydz=\int_{-10}^{-7}(\frac{(8sin(3z)e^{(y)})}{1683})dz|_{7}^{9}\\&\int_{-10}^{-7}(-\frac{(8sin(3z)\cdot(e^{(7)} - e^{(9)}))}{1683})dz=-\frac{(8cos(3z)e^{(7)}\cdot(e^{(2)} - 1))}{5049}|_{-10}^{-7}=7.79\end{align*}$

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Example Question #232 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{11}\int_{-10}^{-6}\int_{-9}^{-5}(\frac{(46z^{2}sin(x + 1))}{(9y)})dxdydz\end{align*}$

Possible Answers:

-532.5

1597

-44.37

-266.2

Correct answer:

-266.2

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{9}^{11}\int_{-10}^{-6}\int_{-9}^{-5}(\frac{(46z^{2}sin(x + 1))}{(9y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{11}\int_{-10}^{-6}\int_{-9}^{-5}(\frac{(46z^{2}sin(x + 1))}{(9y)})dxdydz=\int_{9}^{11}\int_{-10}^{-6}(-\frac{(46z^{2}cos(x + 1))}{(9y)})dydz|_{-9}^{-5}\\&\int_{9}^{11}\int_{-10}^{-6}(-\frac{(46z^{2}\cdot(cos(4) - cos(8)))}{(9y)})dydz=\int_{9}^{11}(-ln(y)\cdot(\frac{(46z^{2}cos(4))}{9}-\frac{ (46z^{2}cos(8))}{9}))dz|_{-10}^{-6}\\&\int_{9}^{11}(-\frac{(46z^{2}ln(\frac{3}{5})\cdot(cos(4) - cos(8)))}{9})dz=-z^{3}\cdot(\frac{(46cos(4)ln(\frac{3}{5}))}{27}-\frac{ (46cos(8)ln(\frac{3}{5}))}{27})|_{9}^{11}=-266.2\end{align*}$

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Example Question #233 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-8}^{-3}\int_{6}^{7}\int_{-9}^{-7}(\frac{(66sin(z + 2)e^{(-\frac{y}{2})})}{(5x)})dxdydz\end{align*}$

Possible Answers:

-0.11

-0.03

-0.05

0.33

Correct answer:

-0.05

Explanation:

$\begin{align*}&\text{Performing a triple integral, order of integration does}\\&\text{not fully matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\end{align*}$

$\begin{align*}\\&\text{Considering our integral:}\\&\int_{-8}^{-3}\int_{6}^{7}\int_{-9}^{-7}(\frac{(66sin(z + 2)e^{(-\frac{y}{2})})}{(5x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-8}^{-3}\int_{6}^{7}\int_{-9}^{-7}(\frac{(66sin(z + 2)e^{(-\frac{y}{2})})}{(5x)})dxdydz=\int_{-8}^{-3}\int_{6}^{7}(\frac{(66sin(z + 2)e^{(-\frac{y}{2})}ln(x))}{5})dydz|_{-9}^{-7}\\&\int_{-8}^{-3}\int_{6}^{7}(\frac{(66sin(z + 2)e^{(-\frac{y}{2})}ln(\frac{7}{9}))}{5})dydz=\int_{-8}^{-3}(-\frac{(132sin(z + 2)e^{(-\frac{y}{2})}ln(\frac{7}{9}))}{5})dz|_{6}^{7}\\&\int_{-8}^{-3}(\frac{(132sin(z + 2)e^{(-\frac{7}{2})}ln(\frac{7}{9})\cdot(e^{(\frac{1}{2})} - 1))}{5})dz=-\frac{(132cos(z + 2)e^{(-\frac{7}{2})}ln(\frac{7}{9})\cdot(e^{(\frac{1}{2})} - 1))}{5}|_{-8}^{-3}=-0.05\end{align*}$

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Example Question #231 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{5}^{10}\int_{10}^{15}\int_{-9}^{-8}(9sin(3y)e^{(\frac{x}{4})})dxdydz\end{align*}$

Possible Answers:

3.33

-0.22

-0.67

Correct answer:

-0.67

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{5}^{10}\int_{10}^{15}\int_{-9}^{-8}(9sin(3y)e^{(\frac{x}{4})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{5}^{10}\int_{10}^{15}\int_{-9}^{-8}(9sin(3y)e^{(\frac{x}{4})})dxdydz=\int_{5}^{10}\int_{10}^{15}(36sin(3y)e^{(\frac{x}{4})})dydz|_{-9}^{-8}\\&\int_{5}^{10}\int_{10}^{15}(36sin(3y)e^{(-\frac{9}{4})}\cdot(e^{(\frac{1}{4})} - 1))dydz=\int_{5}^{10}(-12cos(3y)e^{(-\frac{9}{4})}\cdot(e^{(\frac{1}{4})} - 1))dz|_{10}^{15}\\&\int_{5}^{10}(12e^{(-\frac{9}{4})}\cdot(cos(30) - cos(45))\cdot(e^{(\frac{1}{4})} - 1))dz=12ze^{(-\frac{9}{4})}\cdot(cos(30) - cos(45))\cdot(e^{(\frac{1}{4})} - 1)|_{5}^{10}=-0.67\end{align*}$

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Example Question #232 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-1}\int_{-8}^{-4}\int_{8}^{11}(\frac{(5e^{(-\frac{x}{2})})}{(4^yz^{2})})dxdydz\end{align*}$

Possible Answers:

33500

5584

Correct answer:

5584

Explanation:

$\begin{align*}\\&\text{Considering our integral:}\\&\int_{-6}^{-1}\int_{-8}^{-4}\int_{8}^{11}(\frac{(5e^{(-\frac{x}{2})})}{(4^yz^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-1}\int_{-8}^{-4}\int_{8}^{11}(\frac{(5e^{(-\frac{x}{2})})}{(4^yz^{2})})dxdydz=\int_{-6}^{-1}\int_{-8}^{-4}(-\frac{(10e^{(-\frac{x}{2})})}{(4^{(y)}z^{2})})dydz|_{8}^{11}\\&\int_{-6}^{-1}\int_{-8}^{-4}(\frac{(10e^{(-\frac{11}{2})}\cdot(e^{(\frac{3}{2})} - 1))}{(4^yz^{2})})dydz=\int_{-6}^{-1}(-\frac{(10e^{(-\frac{11}{2})}\cdot(e^{(\frac{3}{2})} - 1))}{(4^{(y)}z^{2}ln(4))})dz|_{-8}^{-4}\\&\int_{-6}^{-1}(\frac{(326400e^{(-\frac{11}{2})}\cdot(e^{(\frac{3}{2})} - 1))}{(z^{2}ln(2))})dz=-\frac{(326400e^{(-\frac{11}{2})}\cdot(e^{(\frac{3}{2})} - 1))}{(zln(2))}|_{-6}^{-1}=5584\end{align*}$

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Example Question #233 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-6}\int_{4}^{5}\int_{10}^{11}(\frac{(6xe^{(-z)})}{(31y)})dxdydz\end{align*}$

Possible Answers:

17466

3492

Correct answer:

3492

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-9}^{-6}\int_{4}^{5}\int_{10}^{11}(\frac{(6xe^{(-z)})}{(31y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-6}\int_{4}^{5}\int_{10}^{11}(\frac{(6xe^{(-z)})}{(31y)})dxdydz=\int_{-9}^{-6}\int_{4}^{5}(\frac{(3x^{2}e^{(-z)})}{(31y)})dydz|_{10}^{11}\\&\int_{-9}^{-6}\int_{4}^{5}(\frac{(63e^{(-z)})}{(31y)})dydz=\int_{-9}^{-6}(\frac{(63e^{(-z)}ln(y))}{31})dz|_{4}^{5}\\&\int_{-9}^{-6}(\frac{(63e^{(-z)}ln(\frac{5}{4}))}{31})dz=-\frac{(63e^{(-z)}ln(\frac{5}{4}))}{31}|_{-9}^{-6}=3492\end{align*}$

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Example Question #234 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{8}\int_{8}^{13}\int_{9}^{12}(\frac{(26cos(y + 1)sin(3z))}{(7x)})dxdydz\end{align*}$

Possible Answers:

-0.28

1.38

-0.83

0.09

Correct answer:

-0.28

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{3}^{8}\int_{8}^{13}\int_{9}^{12}(\frac{(26cos(y + 1)sin(3z))}{(7x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{8}\int_{8}^{13}\int_{9}^{12}(\frac{(26cos(y + 1)sin(3z))}{(7x)})dxdydz=\int_{3}^{8}\int_{8}^{13}(\frac{(26cos(y + 1)sin(3z)ln(x))}{7})dydz|_{9}^{12}\\&\int_{3}^{8}\int_{8}^{13}(\frac{(26cos(y + 1)sin(3z)ln(\frac{4}{3}))}{7})dydz=\int_{3}^{8}(\frac{(26sin(y + 1)sin(3z)ln(\frac{4}{3}))}{7})dz|_{8}^{13}\\&\int_{3}^{8}(-\frac{(26sin(3z)ln(\frac{4}{3})\cdot(sin(9) - sin(14)))}{7})dz=\frac{(26cos(3z)ln(\frac{4}{3})\cdot(sin(9) - sin(14)))}{21}|_{3}^{8}=-0.28\end{align*}$

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Example Question #235 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{11}\int_{-6}^{-2}\int_{-8}^{-4}(\frac{(57cos(z + 2)sin(4x))}{(8y^{2})})dxdydz\end{align*}$

Possible Answers:

1.02

3.05

-6.11

-0.25

Correct answer:

1.02

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{10}^{11}\int_{-6}^{-2}\int_{-8}^{-4}(\frac{(57cos(z + 2)sin(4x))}{(8y^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{11}\int_{-6}^{-2}\int_{-8}^{-4}(\frac{(57cos(z + 2)sin(4x))}{(8y^{2})})dxdydz=\int_{10}^{11}\int_{-6}^{-2}(-\frac{(57cos(4x)cos(z + 2))}{(32y^{2})})dydz|_{-8}^{-4}\\&\int_{10}^{11}\int_{-6}^{-2}(-\frac{(57cos(z + 2)\cdot(\frac{cos(16)}{4}-\frac{ cos(32)}{4}))}{(8y^{2})})dydz=\int_{10}^{11}(\frac{(\frac{(57cos(z + 2)cos(16))}{32}-\frac{ (57cos(z + 2)cos(32))}{32})}{y})dz|_{-6}^{-2}\\&\int_{10}^{11}(-\frac{(19cos(z + 2)\cdot(cos(16) - cos(32)))}{32})dz=-sin(z + 2)\cdot(\frac{(19cos(16))}{32}-\frac{ (19cos(32))}{32})|_{10}^{11}=1.02\end{align*}$

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Example Question #236 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{10}\int_{-9}^{-7}\int_{-6}^{-1}(\frac{(2cos(y + 2))}{11})dxdydz\end{align*}$

Possible Answers:

-8.81

4.41

-1.47

26.44

Correct answer:

4.41

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{7}^{10}\int_{-9}^{-7}\int_{-6}^{-1}(\frac{(2cos(y + 2))}{11})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{10}\int_{-9}^{-7}\int_{-6}^{-1}(\frac{(2cos(y + 2))}{11})dxdydz=\int_{7}^{10}\int_{-9}^{-7}(\frac{(2xcos(y + 2))}{11})dydz|_{-6}^{-1}\\&\int_{7}^{10}\int_{-9}^{-7}(\frac{(10cos(y + 2))}{11})dydz=\int_{7}^{10}(\frac{(10sin(y + 2))}{11})dz|_{-9}^{-7}\\&\int_{7}^{10}(\frac{(10sin(7))}{11}-\frac{ (10sin(5))}{11})dz=-z\cdot(\frac{(10sin(5))}{11}-\frac{ (10sin(7))}{11})|_{7}^{10}=4.41\end{align*}$

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Example Question #237 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-3}\int_{8}^{12}\int_{6}^{8}(\frac{(16e^{(\frac{x}{3})})}{(9y^{2}z^{2})})dxdydz\end{align*}$

Possible Answers:

-1.3

0.26

0.78

0.06

Correct answer:

0.26

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-6}^{-3}\int_{8}^{12}\int_{6}^{8}(\frac{(16e^{(\frac{x}{3})})}{(9y^{2}z^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-3}\int_{8}^{12}\int_{6}^{8}(\frac{(16e^{(\frac{x}{3})})}{(9y^{2}z^{2})})dxdydz=\int_{-6}^{-3}\int_{8}^{12}(\frac{(16e^{(\frac{x}{3})})}{(3y^{2}z^{2})})dydz|_{6}^{8}\\&\int_{-6}^{-3}\int_{8}^{12}(\frac{(16e^{(2)}\cdot(e^{(\frac{2}{3})} - 1))}{(3y^{2}z^{2})})dydz=\int_{-6}^{-3}(-\frac{(16e^{(2)}\cdot(e^{(\frac{2}{3})} - 1))}{(3yz^{2})})dz|_{8}^{12}\\&\int_{-6}^{-3}(\frac{(2e^{(2)}\cdot(e^{(\frac{2}{3})} - 1))}{(9z^{2})})dz=-\frac{(2e^{(2)}\cdot(e^{(\frac{2}{3})} - 1))}{(9z)}|_{-6}^{-3}=0.26\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #231 : Triple Integrals

Example Question #232 : Triple Integrals

Example Question #233 : Triple Integrals

Example Question #231 : Multiple Integration

Example Question #232 : Multiple Integration

Example Question #233 : Multiple Integration

Example Question #234 : Multiple Integration

Example Question #235 : Multiple Integration

Example Question #236 : Multiple Integration

Example Question #237 : Multiple Integration

All Calculus 3 Resources

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