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Calculus 3 : Calculus 3

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Example Questions

Example Question #271 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{13}\int_{-6}^{-3.5}\int_{-4.5}^{-1}(\frac{e^{(-x)}}{(21yz)})dxdydz\end{align*}$

Possible Answers:

-0.59

2.35

0.59

-2.35

Correct answer:

-0.59

Explanation:

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

$\begin{align*}&\int_{10}^{13}\int_{-6}^{-3.5}\int_{-4.5}^{-1}(\frac{e^{(-x)}}{(21yz)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{13}\int_{-6}^{-3.5}\int_{-4.5}^{-1}(\frac{e^{(-x)}}{(21yz)})dxdydz=\int_{10}^{13}\int_{-6}^{-3.5}(-\frac{e^{(-x)}}{(21yz)})dydz|_{-4.5}^{-1}\\&\int_{10}^{13}\int_{-6}^{-3.5}(-\frac{(e^{(1)} - e^{(\frac{9}{2})})}{(21yz)})dydz=\int_{10}^{13}(\frac{(e^{(1)}ln(y)\cdot (e^{(\frac{7}{2})} - 1))}{(21z)})dz|_{-6}^{-3.5}\\&\int_{10}^{13}(\frac{(e^{(1)}ln(\frac{7}{12})\cdot (e^{(\frac{7}{2})} - 1))}{(21z)})dz=\frac{(e^{(1)}ln(\frac{7}{12})ln(z)\cdot (e^{(\frac{7}{2})} - 1))}{21}|_{10}^{13}=-0.59\end{align*}$

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Example Question #404 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{12}\int_{3.5}^{6.5}\int_{-10}^{-5}(\frac{(8cos(3x)e^{(y)})}{z})dxdydz\end{align*}$

Possible Answers:

-100.7

-503.4

2014

125.9

Correct answer:

-503.4

Explanation:

$\begin{align*}&\int_{10}^{12}\int_{3.5}^{6.5}\int_{-10}^{-5}(\frac{(8cos(3x)e^{(y)})}{z})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{12}\int_{3.5}^{6.5}\int_{-10}^{-5}(\frac{(8cos(3x)e^{(y)})}{z})dxdydz=\int_{10}^{12}\int_{3.5}^{6.5}(\frac{(8sin(3x)e^{(y)})}{(3z)})dydz|_{-10}^{-5}\\&\int_{10}^{12}\int_{3.5}^{6.5}(-\frac{(8e^{(y)}\cdot (sin(15) - sin(30)))}{(3z)})dydz=\int_{10}^{12}(-\frac{(8e^{(y)}\cdot (sin(15) - sin(30)))}{(3z)})dz|_{3.5}^{6.5}\\&\int_{10}^{12}(-\frac{(8e^{(\frac{7}{2})}\cdot (sin(15) - sin(30))\cdot (e^{(3)} - 1))}{(3z)})dz=ln(z)\cdot (\frac{(8e^{(\frac{7}{2})}sin(15))}{3}-\frac{ (8e^{(\frac{13}{2})}sin(15))}{3}-\frac{ (8e^{(\frac{7}{2})}sin(30))}{3}+\frac{ (8e^{(\frac{13}{2})}sin(30))}{3})|_{10}^{12}=-503.4\end{align*}$

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Example Question #272 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{12}\int_{3.5}^{4.5}\int_{4.5}^{8}(\frac{(12\cdot 3^{(\frac{x}{4})}\cdot 3^{(\frac{y}{2})})}{(7z^{2})})dxdydz\end{align*}$

Possible Answers:

-17.57

-2.93

8.78

1.76

Correct answer:

8.78

Explanation:

$\begin{align*}&\int_{9}^{12}\int_{3.5}^{4.5}\int_{4.5}^{8}(\frac{(12\cdot 3^{(\frac{x}{4})}\cdot 3^{(\frac{y}{2})})}{(7z^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{12}\int_{3.5}^{4.5}\int_{4.5}^{8}(\frac{(12\cdot 3^{(\frac{x}{4})}\cdot 3^{(\frac{y}{2})})}{(7z^{2})})dxdydz=\int_{9}^{12}\int_{3.5}^{4.5}(\frac{(48\cdot 3^{(\frac{x}{4}+\frac{ y}{2})})}{(7z^{2}ln(3))})dydz|_{4.5}^{8}\\&\int_{9}^{12}\int_{3.5}^{4.5}(-\frac{(144\cdot 3^{(\frac{y}{2})}\cdot (3^{(\frac{1}{8})} - 3))}{(7z^{2}ln(3))})dydz=\int_{9}^{12}(-\frac{(288\cdot 3^{(\frac{y}{2})}\cdot (3^{(\frac{1}{8})} - 3))}{(7z^{2}ln(3)^{2})})dz|_{3.5}^{4.5}\\&\int_{9}^{12}(\frac{(7776\cdot 3^{(\frac{1}{4})} - 2592\cdot 3^{(\frac{3}{4})} - 2592\cdot 3^{(\frac{3}{8})} + 864\cdot 3^{(\frac{7}{8})})}{(7z^{2}ln(3)^{2})})dz=-\frac{(7776\cdot 3^{(\frac{1}{4})} - 2592\cdot 27^{(\frac{1}{4})} - 2592\cdot 27^{(\frac{1}{8})} + 864\cdot 2187^{(\frac{1}{8})})}{(7zln(3)^{2})}|_{9}^{12}=8.78\end{align*}$

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Example Question #401 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{15}\int_{5}^{9.5}\int_{6}^{7.5}(\frac{(71e^{(y)})}{(6xz^{3})})dxdydz\end{align*}$

Possible Answers:

-193.8

484.5

-19.38

96.9

Correct answer:

96.9

Explanation:

$\begin{align*}&\int_{10}^{15}\int_{5}^{9.5}\int_{6}^{7.5}(\frac{(71e^{(y)})}{(6xz^{3})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{15}\int_{5}^{9.5}\int_{6}^{7.5}(\frac{(71e^{(y)})}{(6xz^{3})})dxdydz=\int_{10}^{15}\int_{5}^{9.5}(\frac{(71e^{(y)}ln(x))}{(6z^{3})})dydz|_{6}^{7.5}\\&\int_{10}^{15}\int_{5}^{9.5}(\frac{(71e^{(y)}ln(\frac{5}{4}))}{(6z^{3})})dydz=\int_{10}^{15}(\frac{(71e^{(y)}ln(\frac{5}{4}))}{(6z^{3})})dz|_{5}^{9.5}\\&\int_{10}^{15}(\frac{(71e^{(5)}ln(\frac{5}{4})\cdot (e^{(\frac{9}{2})} - 1))}{(6z^{3})})dz=-\frac{(71e^{(5)}ln(\frac{5}{4})\cdot (e^{(\frac{9}{2})} - 1))}{(12z^{2})}|_{10}^{15}=96.9\end{align*}$

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Example Question #407 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-4.5}\int_{8}^{9.5}\int_{-4.5}^{-2.5}(\frac{29}{(2\cdot 2^xyz^{3})})dxdydz\end{align*}$

Possible Answers:

-0.44

-0.88

1.77

0.22

Correct answer:

-0.88

Explanation:

$\begin{align*}&\int_{-7}^{-4.5}\int_{8}^{9.5}\int_{-4.5}^{-2.5}(\frac{29}{(2\cdot 2^xyz^{3})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-4.5}\int_{8}^{9.5}\int_{-4.5}^{-2.5}(\frac{29}{(2\cdot 2^xyz^{3})})dxdydz=\int_{-7}^{-4.5}\int_{8}^{9.5}(-\frac{29}{(2\cdot 2^xyz^{3}ln(2))})dydz|_{-4.5}^{-2.5}\\&\int_{-7}^{-4.5}\int_{8}^{9.5}(\frac{(174\cdot 2^{(\frac{1}{2})})}{(yz^{3}ln(2))})dydz=\int_{-7}^{-4.5}(\frac{(174\cdot 2^{(\frac{1}{2})}ln(y))}{(z^{3}ln(2))})dz|_{8}^{9.5}\\&\int_{-7}^{-4.5}(\frac{(174\cdot 2^{(\frac{1}{2})}ln(\frac{19}{16}))}{(z^{3}ln(2))})dz=-\frac{(87\cdot 2^{(\frac{1}{2})}ln(\frac{19}{16}))}{(z^{2}ln(2))}|_{-7}^{-4.5}=-0.88\end{align*}$

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Example Question #402 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-8}^{-6.5}\int_{-3.5}^{0.5}\int_{-10}^{-5}(\frac{(76\cdot 3^y)}{(27xz^{2})})dxdydz\end{align*}$

Possible Answers:

0.26

0.01

-0.09

-0.18

Correct answer:

-0.09

Explanation:

$\begin{align*}&\int_{-8}^{-6.5}\int_{-3.5}^{0.5}\int_{-10}^{-5}(\frac{(76\cdot 3^y)}{(27xz^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-8}^{-6.5}\int_{-3.5}^{0.5}\int_{-10}^{-5}(\frac{(76\cdot 3^y)}{(27xz^{2})})dxdydz=\int_{-8}^{-6.5}\int_{-3.5}^{0.5}(\frac{(76\cdot 3^yln(x))}{(27z^{2})})dydz|_{-10}^{-5}\\&\int_{-8}^{-6.5}\int_{-3.5}^{0.5}(-\frac{(76\cdot 3^yln(2))}{(27z^{2})})dydz=\int_{-8}^{-6.5}(-\frac{(76\cdot 3^yln(2))}{(27z^{2}ln(3))})dz|_{-3.5}^{0.5}\\&\int_{-8}^{-6.5}(-\frac{(6080\cdot 3^{(\frac{1}{2})}ln(2))}{(2187z^{2}ln(3))})dz=\frac{(6080\cdot 3^{(\frac{1}{2})}ln(2))}{(2187zln(3))}|_{-8}^{-6.5}=-0.09\end{align*}$

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Example Question #409 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-4}\int_{3}^{8}\int_{3}^{6.5}(\frac{(11\cdot 3^{(\frac{y}{2})}e^{(-x)})}{(180z)})dxdydz\end{align*}$

Possible Answers:

0.06

-0.23

-0.46

0.68

Correct answer:

-0.23

Explanation:

$\begin{align*}&\int_{-7}^{-4}\int_{3}^{8}\int_{3}^{6.5}(\frac{(11\cdot 3^{(\frac{y}{2})}e^{(-x)})}{(180z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-4}\int_{3}^{8}\int_{3}^{6.5}(\frac{(11\cdot 3^{(\frac{y}{2})}e^{(-x)})}{(180z)})dxdydz=\int_{-7}^{-4}\int_{3}^{8}(-\frac{(11\cdot 3^{(\frac{y}{2})}e^{(-x)})}{(180z)})dydz|_{3}^{6.5}\\&\int_{-7}^{-4}\int_{3}^{8}(\frac{(11\cdot 3^{(\frac{y}{2})}\cdot (e^{(-3)} - e^{(-\frac{13}{2})}))}{(180z)})dydz=\int_{-7}^{-4}(\frac{(11\cdot 3^{(\frac{y}{2})}e^{(-\frac{13}{2})}\cdot (e^{(\frac{7}{2})} - 1))}{(90zln(3))})dz|_{3}^{8}\\&\int_{-7}^{-4}(\frac{(297e^{(-3)} - 297e^{(-\frac{13}{2})} - 11\cdot 3^{(\frac{1}{2})}e^{(-3)} + 11\cdot 3^{(\frac{1}{2})}e^{(-\frac{13}{2})})}{(30zln(3))})dz=\frac{(e^{(-\frac{13}{2})}ln(z)\cdot (297e^{(\frac{7}{2})} - 11\cdot 3^{(\frac{1}{2})}e^{(\frac{7}{2})} + 11\cdot 3^{(\frac{1}{2})} - 297))}{(30ln(3))}|_{-7}^{-4}=-0.23\end{align*}$

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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:

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

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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:

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

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #271 : Triple Integration In Cartesian Coordinates

Example Question #404 : Multiple Integration

Example Question #272 : Triple Integration In Cartesian Coordinates

Example Question #401 : Triple Integrals

Example Question #407 : Multiple Integration

Example Question #402 : Triple Integrals

Example Question #409 : Multiple Integration

Example Question #411 : Triple Integrals

Example Question #412 : Triple Integrals

Example Question #413 : Triple Integrals

All Calculus 3 Resources

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