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Calculus 3 : Multiple Integration

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Example Questions

Example Question #218 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-8}\int_{8}^{11.5}\int_{-6}^{-3}(\frac{(72cos(x + 2)sin(4z))}{(11y^{2})})dxdydz\end{align*}$

Possible Answers:

-0.02

-0.29

0.03

0.1

Correct answer:

0.1

Explanation:

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

$\begin{align*}&\int_{-9}^{-8}\int_{8}^{11.5}\int_{-6}^{-3}(\frac{(72cos(x + 2)sin(4z))}{(11y^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-8}\int_{8}^{11.5}\int_{-6}^{-3}(\frac{(72cos(x + 2)sin(4z))}{(11y^{2})})dxdydz=\int_{-9}^{-8}\int_{8}^{11.5}(\frac{(72sin(x + 2)sin(4z))}{(11y^{2})})dydz|_{-6}^{-3}\\&\int_{-9}^{-8}\int_{8}^{11.5}(-\frac{(72sin(4z)\cdot (sin(1) - sin(4)))}{(11y^{2})})dydz=\int_{-9}^{-8}(\frac{(\frac{(72sin(4z)sin(1))}{11}-\frac{ (72sin(4z)sin(4))}{11})}{y})dz|_{8}^{11.5}\\&\int_{-9}^{-8}(-\frac{(63sin(4z)\cdot (sin(1) - sin(4)))}{253})dz=cos(4z)\cdot (\frac{(63sin(1))}{1012}-\frac{ (63sin(4))}{1012})|_{-9}^{-8}=0.1\end{align*}$

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Example Question #219 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3.5}^{1.75}\int_{-6}^{-4}\int_{6}^{9.5}(10cos(4y)sin(x + 1)e^{(2z)})dxdydz\end{align*}$

Possible Answers:

-188.6

-31.43

10.48

125.7

Correct answer:

-31.43

Explanation:

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

$\begin{align*}&\int_{-3.5}^{1.75}\int_{-6}^{-4}\int_{6}^{9.5}(10cos(4y)sin(x + 1)e^{(2z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3.5}^{1.75}\int_{-6}^{-4}\int_{6}^{9.5}(10cos(4y)sin(x + 1)e^{(2z)})dxdydz=\int_{-3.5}^{1.75}\int_{-6}^{-4}(-10cos(x + 1)cos(4y)e^{(2z)})dydz|_{6}^{9.5}\\&\int_{-3.5}^{1.75}\int_{-6}^{-4}(10cos(4y)e^{(2z)}\cdot (cos(7) - cos(\frac{21}{2})))dydz=\int_{-3.5}^{1.75}(\frac{(5sin(4y)e^{(2z)}\cdot (cos(7) - cos(\frac{21}{2})))}{2})dz|_{-6}^{-4}\\&\int_{-3.5}^{1.75}(-\frac{(5e^{(2z)}\cdot (cos(7) - cos(\frac{21}{2}))\cdot (sin(16) - sin(24)))}{2})dz=-\frac{(5e^{(2z)}\cdot (cos(7) - cos(\frac{21}{2}))\cdot (sin(16) - sin(24)))}{4}|_{-3.5}^{1.75}=-31.43\end{align*}$

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Example Question #220 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{6.5}\int_{-8}^{-7}\int_{-4}^{1}(\frac{(y^{2}e^{(-x)}e^{(z)})}{14})dxdydz\end{align*}$

Possible Answers:

422300

140800

Correct answer:

140800

Explanation:

$\begin{align*}&\int_{3}^{6.5}\int_{-8}^{-7}\int_{-4}^{1}(\frac{(y^{2}e^{(-x)}e^{(z)})}{14})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{6.5}\int_{-8}^{-7}\int_{-4}^{1}(\frac{(y^{2}e^{(-x)}e^{(z)})}{14})dxdydz=\int_{3}^{6.5}\int_{-8}^{-7}(-\frac{(y^{2}e^{(z - x)})}{14})dydz|_{-4}^{1}\\&\int_{3}^{6.5}\int_{-8}^{-7}(\frac{(y^{2}e^{(-1)}e^{(z)}\cdot (e^{(5)} - 1))}{14})dydz=\int_{3}^{6.5}(\frac{(y^{3}e^{(-1)}e^{(z)}\cdot (e^{(5)} - 1))}{42})dz|_{-8}^{-7}\\&\int_{3}^{6.5}(\frac{(169e^{(-1)}e^{(z)}\cdot (e^{(5)} - 1))}{42})dz=e^{(z - 1)}\cdot (\frac{(169e^{(5)})}{42}-\frac{ 169}{42})|_{3}^{6.5}=140800\end{align*}$

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Example Question #831 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{11}\int_{4}^{7}\int_{-9}^{-7}(\frac{(2cos(z + 2)e^{(2y)})}{(13x^{2})})dxdydz\end{align*}$

Possible Answers:

16644

4160

Correct answer:

4160

Explanation:

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

$\begin{align*}&\int_{9}^{11}\int_{4}^{7}\int_{-9}^{-7}(\frac{(2cos(z + 2)e^{(2y)})}{(13x^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{11}\int_{4}^{7}\int_{-9}^{-7}(\frac{(2cos(z + 2)e^{(2y)})}{(13x^{2})})dxdydz=\int_{9}^{11}\int_{4}^{7}(-\frac{(2cos(z + 2)e^{(2y)})}{(13x)})dydz|_{-9}^{-7}\\&\int_{9}^{11}\int_{4}^{7}(\frac{(4cos(z + 2)e^{(2y)})}{819})dydz=\int_{9}^{11}(\frac{(2cos(z + 2)e^{(2y)})}{819})dz|_{4}^{7}\\&\int_{9}^{11}(\frac{(2cos(z + 2)e^{(8)}\cdot (e^{(6)} - 1))}{819})dz=\frac{(2sin(z + 2)e^{(8)}\cdot (e^{(6)} - 1))}{819}|_{9}^{11}=4160\end{align*}$

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Example Question #221 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{11}^{13}\int_{-3.5}^{-1.5}\int_{-4}^{1}(\frac{(11\cdot 2^{(\frac{x}{4})}e^{(-2y)})}{(8z)})dxdydz\end{align*}$

Possible Answers:

-98.35

122.9

491.8

Correct answer:

491.8

Explanation:

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

$\begin{align*}&\int_{11}^{13}\int_{-3.5}^{-1.5}\int_{-4}^{1}(\frac{(11\cdot 2^{(\frac{x}{4})}e^{(-2y)})}{(8z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{11}^{13}\int_{-3.5}^{-1.5}\int_{-4}^{1}(\frac{(11\cdot 2^{(\frac{x}{4})}e^{(-2y)})}{(8z)})dxdydz=\int_{11}^{13}\int_{-3.5}^{-1.5}(\frac{(11\cdot 2^{(\frac{x}{4})}e^{(-2y)})}{(2zln(2))})dydz|_{-4}^{1}\\&\int_{11}^{13}\int_{-3.5}^{-1.5}(\frac{(11e^{(-2y)}\cdot (2\cdot 2^{(\frac{1}{4})} - 1))}{(4zln(2))})dydz=\int_{11}^{13}(-\frac{(11e^{(-2y)}\cdot (2\cdot 2^{(\frac{1}{4})} - 1))}{(8zln(2))})dz|_{-3.5}^{-1.5}\\&\int_{11}^{13}(\frac{(11e^{(3)} - 11e^{(7)} - 22\cdot 2^{(\frac{1}{4})}e^{(3)} + 22\cdot 2^{(\frac{1}{4})}e^{(7)})}{(8zln(2))})dz=\frac{(ln(z)\cdot (11e^{(3)} - 11e^{(7)} - 22\cdot 2^{(\frac{1}{4})}e^{(3)} + 22\cdot 2^{(\frac{1}{4})}e^{(7)}))}{(8ln(2))}|_{11}^{13}=491.8\end{align*}$

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Example Question #831 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-7.5}\int_{-4.5}^{-2.5}\int_{4}^{5}(\frac{(3^{(\frac{x}{4})}cos(z + 2))}{(26\cdot 3^y)})dxdydz\end{align*}$

Possible Answers:

20.54

102.7

-4.11

-123.2

Correct answer:

20.54

Explanation:

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

$\begin{align*}&\int_{-9}^{-7.5}\int_{-4.5}^{-2.5}\int_{4}^{5}(\frac{(3^{(\frac{x}{4})}cos(z + 2))}{(26\cdot 3^y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-7.5}\int_{-4.5}^{-2.5}\int_{4}^{5}(\frac{(3^{(\frac{x}{4})}cos(z + 2))}{(26\cdot 3^y)})dxdydz=\int_{-9}^{-7.5}\int_{-4.5}^{-2.5}(\frac{(2\cdot (\frac{1}{3})^y\cdot 3^{(\frac{x}{4})}cos(z + 2))}{(13ln(3))})dydz|_{4}^{5}\\&\int_{-9}^{-7.5}\int_{-4.5}^{-2.5}(\frac{(6cos(z + 2)\cdot (3^{(\frac{1}{4})} - 1))}{(13\cdot 3^yln(3))})dydz=\int_{-9}^{-7.5}(\frac{(6\cdot (cos(z + 2) - 3^{(\frac{1}{4})}cos(z + 2)))}{(13\cdot 3^yln(3)^{2})})dz|_{-4.5}^{-2.5}\\&\int_{-9}^{-7.5}(-\frac{(432\cdot 3^{(\frac{1}{2})}cos(z + 2) - 432\cdot 3^{(\frac{3}{4})}cos(z + 2))}{(13ln(3)^{2})})dz=-\frac{(432\cdot 3^{(\frac{1}{2})}sin(z + 2) - 432\cdot 27^{(\frac{1}{4})}sin(z + 2))}{(13ln(3)^{2})}|_{-9}^{-7.5}=20.54\end{align*}$

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Example Question #831 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{-4}^{-1.5}\int_{-4}^{0.5}\int_{-9}^{-4.5}(\frac{(43x^{2}e^{(y)}e^{(z)})}{12})dxdydz\end{align*}$

Possible Answers:

-42.4

254.4

1272

Correct answer:

254.4

Explanation:

$\begin{align*}&\int_{-4}^{-1.5}\int_{-4}^{0.5}\int_{-9}^{-4.5}(\frac{(43x^{2}e^{(y)}e^{(z)})}{12})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4}^{-1.5}\int_{-4}^{0.5}\int_{-9}^{-4.5}(\frac{(43x^{2}e^{(y)}e^{(z)})}{12})dxdydz=\int_{-4}^{-1.5}\int_{-4}^{0.5}(\frac{(43x^{3}e^{(y + z)})}{36})dydz|_{-9}^{-4.5}\\&\int_{-4}^{-1.5}\int_{-4}^{0.5}(\frac{(24381e^{(y + z)})}{32})dydz=\int_{-4}^{-1.5}(\frac{(24381e^{(y + z)})}{32})dz|_{-4}^{0.5}\\&\int_{-4}^{-1.5}(\frac{(24381e^{(-4)}e^{(z)}\cdot (e^{(\frac{9}{2})} - 1))}{32})dz=e^{(z - 4)}\cdot (\frac{(24381e^{(\frac{9}{2})})}{32}-\frac{ 24381}{32})|_{-4}^{-1.5}=254.4\end{align*}$

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Example Question #358 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{5.5}\int_{4}^{5}\int_{-6}^{-1.5}(\frac{(3\cdot 3^{(\frac{z}{2})}\cdot 3^y)}{(74x^{2})})dxdydz\end{align*}$

Possible Answers:

-333.4

83.36

250.1

-16.67

Correct answer:

83.36

Explanation:

$\begin{align*}&\int_{3}^{5.5}\int_{4}^{5}\int_{-6}^{-1.5}(\frac{(3\cdot 3^{(\frac{z}{2})}\cdot 3^y)}{(74x^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{5.5}\int_{4}^{5}\int_{-6}^{-1.5}(\frac{(3\cdot 3^{(\frac{z}{2})}\cdot 3^y)}{(74x^{2})})dxdydz=\int_{3}^{5.5}\int_{4}^{5}(-\frac{(3\cdot 3^{(y +\frac{ z}{2})})}{(74x)})dydz|_{-6}^{-1.5}\\&\int_{3}^{5.5}\int_{4}^{5}(\frac{(3\cdot 3^{(y +\frac{ z}{2})})}{148})dydz=\int_{3}^{5.5}(\frac{(3\cdot 3^{(y +\frac{ z}{2})})}{(148ln(3))})dz|_{4}^{5}\\&\int_{3}^{5.5}(\frac{(243\cdot (3^{(\frac{1}{2})})^z)}{(74ln(3))})dz=\frac{(243\cdot 3^{(\frac{z}{2})})}{(37ln(3)^{2})}|_{3}^{5.5}=83.36\end{align*}$

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Example Question #359 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-7}\int_{-6}^{-3.5}\int_{6}^{11}(\frac{cos(x + 1)}{78})dxdydz\end{align*}$

Possible Answers:

-0.08

-0.31

0.02

0.08

Correct answer:

-0.08

Explanation:

$\begin{align*}&\int_{-9}^{-7}\int_{-6}^{-3.5}\int_{6}^{11}(\frac{cos(x + 1)}{78})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-7}\int_{-6}^{-3.5}\int_{6}^{11}(\frac{cos(x + 1)}{78})dxdydz=\int_{-9}^{-7}\int_{-6}^{-3.5}(\frac{sin(x + 1)}{78})dydz|_{6}^{11}\\&\int_{-9}^{-7}\int_{-6}^{-3.5}(\frac{sin(12)}{78}-\frac{ sin(7)}{78})dydz=\int_{-9}^{-7}(-y\cdot (\frac{sin(7)}{78}-\frac{ sin(12)}{78}))dz|_{-6}^{-3.5}\\&\int_{-9}^{-7}(\frac{(5sin(12))}{156}-\frac{ (5sin(7))}{156})dz=-z\cdot (\frac{(5sin(7))}{156}-\frac{ (5sin(12))}{156})|_{-9}^{-7}=-0.08\end{align*}$

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Example Question #360 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{4.5}^{6.5}\int_{-5}^{-1.5}\int_{4}^{7.5}(\frac{(41\cdot 3^z)}{(4\cdot 3^{(\frac{x}{2})}\cdot 3^{(\frac{y}{2})})})dxdydz\end{align*}$

Possible Answers:

14611

43822

Correct answer:

43822

Explanation:

$\begin{align*}&\int_{4.5}^{6.5}\int_{-5}^{-1.5}\int_{4}^{7.5}(\frac{(41\cdot 3^z)}{(4\cdot 3^{(\frac{x}{2})}\cdot 3^{(\frac{y}{2})})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4.5}^{6.5}\int_{-5}^{-1.5}\int_{4}^{7.5}(\frac{(41\cdot 3^z)}{(4\cdot 3^{(\frac{x}{2})}\cdot 3^{(\frac{y}{2})})})dxdydz=\int_{4.5}^{6.5}\int_{-5}^{-1.5}(-\frac{(41\cdot 3^z)}{(2\cdot 3^{(\frac{x}{2})}\cdot 3^{(\frac{y}{2})}ln(3))})dydz|_{4}^{7.5}\\&\int_{4.5}^{6.5}\int_{-5}^{-1.5}(-\frac{(3^{(z -\frac{ y}{2})}\cdot (\frac{(41\cdot 3^{(\frac{1}{4})})}{162}-\frac{ 41}{18}))}{ln(3)})dydz=\int_{4.5}^{6.5}(\frac{(41\cdot 3^{(z -\frac{ y}{2})}\cdot (3^{(\frac{1}{4})} - 9))}{(81ln(3)^{2})})dz|_{-5}^{-1.5}\\&\int_{4.5}^{6.5}(\frac{(41\cdot 3^z\cdot (27\cdot 3^{(\frac{1}{2})} - 6\cdot 3^{(\frac{3}{4})} + 1))}{(27ln(3)^{2})})dz=\frac{(41\cdot 3^z\cdot (27\cdot 3^{(\frac{1}{2})} - 6\cdot 3^{(\frac{3}{4})} + 1))}{(27ln(3)^{3})}|_{4.5}^{6.5}=43822\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #218 : Triple Integration In Cartesian Coordinates

Example Question #219 : Triple Integration In Cartesian Coordinates

Example Question #220 : Triple Integration In Cartesian Coordinates

Example Question #831 : Calculus 3

Example Question #221 : Triple Integration In Cartesian Coordinates

Example Question #831 : Calculus 3

Example Question #831 : Calculus 3

Example Question #358 : Multiple Integration

Example Question #359 : Multiple Integration

Example Question #360 : Multiple Integration

All Calculus 3 Resources

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