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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #272 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-5.5}\int_{6}^{8}\int_{5}^{9}(\frac{(2^{(\frac{x}{3})}sin(z + 1))}{(6y)})dxdydz\end{align*}$

Possible Answers:

0.2

1.17

-1.17

-0.29

Correct answer:

1.17

Explanation:

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

$\begin{align*}&\int_{-7}^{-5.5}\int_{6}^{8}\int_{5}^{9}(\frac{(2^{(\frac{x}{3})}sin(z + 1))}{(6y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-5.5}\int_{6}^{8}\int_{5}^{9}(\frac{(2^{(\frac{x}{3})}sin(z + 1))}{(6y)})dxdydz=\int_{-7}^{-5.5}\int_{6}^{8}(\frac{(2^{(\frac{x}{3})}sin(z + 1))}{(2yln(2))})dydz|_{5}^{9}\\&\int_{-7}^{-5.5}\int_{6}^{8}(-\frac{(sin(z + 1)\cdot(6\cdot2^{(\frac{2}{3})} - 24))}{(6yln(2))})dydz=\int_{-7}^{-5.5}(\frac{(ln(y)\cdot(4sin(z + 1) - 4^{(\frac{1}{3})}sin(z + 1)))}{ln(2)})dz|_{6}^{8}\\&\int_{-7}^{-5.5}(\frac{(4sin(z + 1)ln(\frac{4}{3}) - 2^{(\frac{2}{3})}sin(z + 1)ln(\frac{4}{3}))}{ln(2)})dz=\frac{(2ln(\frac{4}{3})\cdot(2^{(\frac{2}{3})} - 4))}{(ln(2)\cdot(tan(\frac{z}{2}+\frac{ 1}{2})^{2} + 1))}|_{-7}^{-5.5}=1.17\end{align*}$

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Example Question #273 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-4}\int_{-3}^{1.5}\int_{6}^{8.5}(\frac{(53sin(z + 1))}{(4\cdot3^{(\frac{y}{2})}x)})dxdydz\end{align*}$

Possible Answers:

-203.6

101.8

50.91

-16.97

Correct answer:

50.91

Explanation:

$\begin{align*}&\int_{-6}^{-4}\int_{-3}^{1.5}\int_{6}^{8.5}(\frac{(53sin(z + 1))}{(4\cdot3^{(\frac{y}{2})}x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-4}\int_{-3}^{1.5}\int_{6}^{8.5}(\frac{(53sin(z + 1))}{(4\cdot3^{(\frac{y}{2})}x)})dxdydz=\int_{-6}^{-4}\int_{-3}^{1.5}(\frac{(53sin(z + 1)ln(x))}{(4\cdot3^{(\frac{y}{2})})})dydz|_{6}^{8.5}\\&\int_{-6}^{-4}\int_{-3}^{1.5}(\frac{(53sin(z + 1)ln(\frac{17}{12}))}{(4\cdot3^{(\frac{y}{2})})})dydz=\int_{-6}^{-4}(-\frac{(53sin(z + 1)ln(\frac{17}{12}))}{(2\cdot3^{(\frac{y}{2})}ln(3))})dz|_{-3}^{1.5}\\&\int_{-6}^{-4}(\frac{(53sin(z + 1)ln(\frac{17}{12})\cdot(18\cdot3^{(\frac{1}{2})} - 2\cdot3^{(\frac{1}{4})}))}{(12ln(3))})dz=-\frac{(53cos(z + 1)ln(\frac{17}{12})\cdot(18\cdot3^{(\frac{1}{2})} - 2\cdot3^{(\frac{1}{4})}))}{(12ln(3))}|_{-6}^{-4}=50.91\end{align*}$

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Example Question #274 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{10}\int_{-3}^{-1}\int_{10}^{14.5}(\frac{(51\cdot3^ysin(3x)sin(z + 1))}{10})dxdydz\end{align*}$

Possible Answers:

0.05

-0.02

-0.1

0.1

Correct answer:

0.05

Explanation:

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

$\begin{align*}&\int_{7}^{10}\int_{-3}^{-1}\int_{10}^{14.5}(\frac{(51\cdot3^ysin(3x)sin(z + 1))}{10})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{10}\int_{-3}^{-1}\int_{10}^{14.5}(\frac{(51\cdot3^ysin(3x)sin(z + 1))}{10})dxdydz=\int_{7}^{10}\int_{-3}^{-1}(-\frac{(17\cdot3^ycos(3x)sin(z + 1))}{10})dydz|_{10}^{14.5}\\&\int_{7}^{10}\int_{-3}^{-1}(\frac{(51\cdot3^ysin(z + 1)\cdot(\frac{cos(30)}{3}-\frac{ cos(\frac{87}{2})}{3}))}{10})dydz=\int_{7}^{10}(\frac{(17\cdot3^ysin(z + 1)cos(30) - 17\cdot3^ysin(z + 1)cos(\frac{87}{2}))}{(10ln(3))})dz|_{-3}^{-1}\\&\int_{7}^{10}(\frac{(68sin(z + 1)cos(30) - 68sin(z + 1)cos(\frac{87}{2}))}{(135ln(3))})dz=-\frac{(136cos(30)cos(\frac{z}{2}+\frac{ 1}{2})^{2} - 136cos(\frac{87}{2})cos(\frac{z}{2}+\frac{ 1}{2})^{2})}{(135ln(3))}|_{7}^{10}=0.05\end{align*}$

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Example Question #275 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-1}\int_{-5}^{-2}\int_{4}^{5.5}(\frac{(17sin(z + 2))}{(2\cdot2^yx)})dxdydz\end{align*}$

Possible Answers:

-130.6

65.28

783.3

-21.76

Correct answer:

-130.6

Explanation:

$\begin{align*}&\int_{-6}^{-1}\int_{-5}^{-2}\int_{4}^{5.5}(\frac{(17sin(z + 2))}{(2\cdot2^yx)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-1}\int_{-5}^{-2}\int_{4}^{5.5}(\frac{(17sin(z + 2))}{(2\cdot2^yx)})dxdydz=\int_{-6}^{-1}\int_{-5}^{-2}(\frac{(17\cdot(\frac{1}{2})^ysin(z + 2)ln(x))}{2})dydz|_{4}^{5.5}\\&\int_{-6}^{-1}\int_{-5}^{-2}(\frac{(17sin(z + 2)ln(\frac{11}{8}))}{(2\cdot2^y)})dydz=\int_{-6}^{-1}(-\frac{(17sin(z + 2)ln(\frac{11}{8}))}{(2\cdot2^yln(2))})dz|_{-5}^{-2}\\&\int_{-6}^{-1}(\frac{(238sin(z + 2)ln(\frac{11}{8}))}{ln(2)})dz=-\frac{(238cos(z + 2)ln(\frac{11}{8}))}{ln(2)}|_{-6}^{-1}=-130.6\end{align*}$

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Example Question #276 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{11.5}\int_{-7}^{-6}\int_{8}^{11}(\frac{(5z^{2})}{(41xy^{2})})dxdydz\end{align*}$

Possible Answers:

-0.06

0.98

0.24

-1.22

Correct answer:

0.24

Explanation:

$\begin{align*}&\int_{9}^{11.5}\int_{-7}^{-6}\int_{8}^{11}(\frac{(5z^{2})}{(41xy^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{11.5}\int_{-7}^{-6}\int_{8}^{11}(\frac{(5z^{2})}{(41xy^{2})})dxdydz=\int_{9}^{11.5}\int_{-7}^{-6}(\frac{(5z^{2}ln(x))}{(41y^{2})})dydz|_{8}^{11}\\&\int_{9}^{11.5}\int_{-7}^{-6}(\frac{(5z^{2}ln(\frac{11}{8}))}{(41y^{2})})dydz=\int_{9}^{11.5}(-\frac{(5z^{2}ln(\frac{11}{8}))}{(41y)})dz|_{-7}^{-6}\\&\int_{9}^{11.5}(\frac{(5z^{2}ln(\frac{11}{8}))}{1722})dz=\frac{(5z^{3}ln(\frac{11}{8}))}{5166}|_{9}^{11.5}=0.24\end{align*}$

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Example Question #277 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-5}\int_{3.5}^{8}\int_{6}^{9}(\frac{(2\cdot2^{(\frac{y}{2})}sin(3z))}{13})dxdydz\end{align*}$

Possible Answers:

15.74

-1.31

2.62

-13.11

Correct answer:

2.62

Explanation:

$\begin{align*}&\int_{-9}^{-5}\int_{3.5}^{8}\int_{6}^{9}(\frac{(2\cdot2^{(\frac{y}{2})}sin(3z))}{13})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-5}\int_{3.5}^{8}\int_{6}^{9}(\frac{(2\cdot2^{(\frac{y}{2})}sin(3z))}{13})dxdydz=\int_{-9}^{-5}\int_{3.5}^{8}(\frac{(2\cdot2^{(\frac{y}{2})}xsin(3z))}{13})dydz|_{6}^{9}\\&\int_{-9}^{-5}\int_{3.5}^{8}(\frac{(6\cdot2^{(\frac{y}{2})}sin(3z))}{13})dydz=\int_{-9}^{-5}(\frac{(12\cdot2^{(\frac{y}{2})}sin(3z))}{(13ln(2))})dz|_{3.5}^{8}\\&\int_{-9}^{-5}(-\frac{(6sin(3z)\cdot(4\cdot2^{(\frac{3}{4})} - 32))}{(13ln(2))})dz=\frac{(2cos(3z)\cdot(4\cdot2^{(\frac{3}{4})} - 32))}{(13ln(2))}|_{-9}^{-5}=2.62\end{align*}$

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Example Question #278 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{12}\int_{-8}^{-5}\int_{6}^{11}(\frac{(3x^{2}sin(4y))}{(10z)})dxdydz\end{align*}$

Possible Answers:

3.42

-10.25

0.85

-1.71

Correct answer:

3.42

Explanation:

$\begin{align*}&\int_{9}^{12}\int_{-8}^{-5}\int_{6}^{11}(\frac{(3x^{2}sin(4y))}{(10z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{12}\int_{-8}^{-5}\int_{6}^{11}(\frac{(3x^{2}sin(4y))}{(10z)})dxdydz=\int_{9}^{12}\int_{-8}^{-5}(\frac{(x^{3}sin(4y))}{(10z)})dydz|_{6}^{11}\\&\int_{9}^{12}\int_{-8}^{-5}(\frac{(223sin(4y))}{(2z)})dydz=\int_{9}^{12}(-\frac{(223cos(4y))}{(8z)})dz|_{-8}^{-5}\\&\int_{9}^{12}(-\frac{(223\cdot(\frac{cos(20)}{4}-\frac{ cos(32)}{4}))}{(2z)})dz=-ln(z)\cdot(\frac{(223cos(20))}{8}-\frac{ (223cos(32))}{8})|_{9}^{12}=3.42\end{align*}$

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Example Question #281 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{0.5}\int_{6}^{7.5}\int_{8}^{9}(\frac{(8x^{2}y^{2}e^{(-2z)})}{69})dxdydz\end{align*}$

Possible Answers:

232000

116000

Correct answer:

116000

Explanation:

$\begin{align*}&\int_{-3}^{0.5}\int_{6}^{7.5}\int_{8}^{9}(\frac{(8x^{2}y^{2}e^{(-2z)})}{69})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{0.5}\int_{6}^{7.5}\int_{8}^{9}(\frac{(8x^{2}y^{2}e^{(-2z)})}{69})dxdydz=\int_{-3}^{0.5}\int_{6}^{7.5}(\frac{(8x^{3}y^{2}e^{(-2z)})}{207})dydz|_{8}^{9}\\&\int_{-3}^{0.5}\int_{6}^{7.5}(\frac{(1736y^{2}e^{(-2z)})}{207})dydz=\int_{-3}^{0.5}(\frac{(1736y^{3}e^{(-2z)})}{621})dz|_{6}^{7.5}\\&\int_{-3}^{0.5}(\frac{(13237e^{(-2z)})}{23})dz=-\frac{(13237e^{(-2z)})}{46}|_{-3}^{0.5}=116000\end{align*}$

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Example Question #282 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{3.5}^{7.5}\int_{-6}^{-4.5}\int_{9}^{13}(34xcos(3y)e^{(-2z)})dxdydz\end{align*}$

Possible Answers:

-0.35

0.07

2.12

-0.09

Correct answer:

-0.35

Explanation:

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

$\begin{align*}&\int_{3.5}^{7.5}\int_{-6}^{-4.5}\int_{9}^{13}(34xcos(3y)e^{(-2z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3.5}^{7.5}\int_{-6}^{-4.5}\int_{9}^{13}(34xcos(3y)e^{(-2z)})dxdydz=\int_{3.5}^{7.5}\int_{-6}^{-4.5}(17x^{2}cos(3y)e^{(-2z)})dydz|_{9}^{13}\\&\int_{3.5}^{7.5}\int_{-6}^{-4.5}(1496cos(3y)e^{(-2z)})dydz=\int_{3.5}^{7.5}(\frac{(1496sin(3y)e^{(-2z)})}{3})dz|_{-6}^{-4.5}\\&\int_{3.5}^{7.5}(\frac{(1496e^{(-2z)}\cdot(sin(18) - sin(\frac{27}{2})))}{3})dz=-\frac{(748e^{(-2z)}\cdot(sin(18) - sin(\frac{27}{2})))}{3}|_{3.5}^{7.5}=-0.35\end{align*}$

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Example Question #283 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-5}^{-2}\int_{-4}^{1}\int_{9}^{12.5}(\frac{(11e^{(-2y)}e^{(z)})}{28})dxdydz\end{align*}$

Possible Answers:

1581

-790.6

263.5

-52.71

Correct answer:

263.5

Explanation:

$\begin{align*}&\int_{-5}^{-2}\int_{-4}^{1}\int_{9}^{12.5}(\frac{(11e^{(-2y)}e^{(z)})}{28})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-5}^{-2}\int_{-4}^{1}\int_{9}^{12.5}(\frac{(11e^{(-2y)}e^{(z)})}{28})dxdydz=\int_{-5}^{-2}\int_{-4}^{1}(\frac{(11xe^{(-2y)}e^{(z)})}{28})dydz|_{9}^{12.5}\\&\int_{-5}^{-2}\int_{-4}^{1}(\frac{(11e^{(z - 2y)})}{8})dydz=\int_{-5}^{-2}(-\frac{(11e^{(z - 2y)})}{16})dz|_{-4}^{1}\\&\int_{-5}^{-2}(\frac{(11e^{(-2)}e^{(z)}\cdot(e^{(10)} - 1))}{16})dz=e^{(z - 2)}\cdot(\frac{(11e^{(10)})}{16}-\frac{ 11}{16})|_{-5}^{-2}=263.5\end{align*}$

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Calculus 3 : Triple Integration in Cartesian Coordinates

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #272 : Multiple Integration

Example Question #273 : Multiple Integration

Example Question #274 : Multiple Integration

Example Question #275 : Multiple Integration

Example Question #276 : Multiple Integration

Example Question #277 : Multiple Integration

Example Question #278 : Multiple Integration

Example Question #281 : Multiple Integration

Example Question #282 : Multiple Integration

Example Question #283 : Multiple Integration

All Calculus 3 Resources

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