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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #791 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{11.5}\int_{10}^{13.5}\int_{6}^{9}(\frac{(11sin(x + 1))}{35})dxdydz\end{align*}$

Possible Answers:

4.38

-8.76

13.14

-1.46

Correct answer:

4.38

Explanation:

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

$\begin{align*}&\int_{9}^{11.5}\int_{10}^{13.5}\int_{6}^{9}(\frac{(11sin(x + 1))}{35})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{11.5}\int_{10}^{13.5}\int_{6}^{9}(\frac{(11sin(x + 1))}{35})dxdydz=\int_{9}^{11.5}\int_{10}^{13.5}(-\frac{(11cos(x + 1))}{35})dydz|_{6}^{9}\\&\int_{9}^{11.5}\int_{10}^{13.5}(\frac{(11cos(7))}{35}-\frac{ (11cos(10))}{35})dydz=\int_{9}^{11.5}(y\cdot(\frac{(11cos(7))}{35}-\frac{ (11cos(10))}{35}))dz|_{10}^{13.5}\\&\int_{9}^{11.5}(\frac{(11cos(7))}{10}-\frac{ (11cos(10))}{10})dz=z\cdot(\frac{(11cos(7))}{10}-\frac{ (11cos(10))}{10})|_{9}^{11.5}=4.38\end{align*}$

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Example Question #311 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-4.5}^{-3.5}\int_{5}^{7.5}\int_{9}^{11.5}(\frac{(34e^{(-y)}e^{(-z)})}{(3x)})dxdydz\end{align*}$

Possible Answers:

0.16

-3.91

-0.49

0.98

Correct answer:

0.98

Explanation:

$\begin{align*}&\int_{-4.5}^{-3.5}\int_{5}^{7.5}\int_{9}^{11.5}(\frac{(34e^{(-y)}e^{(-z)})}{(3x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4.5}^{-3.5}\int_{5}^{7.5}\int_{9}^{11.5}(\frac{(34e^{(-y)}e^{(-z)})}{(3x)})dxdydz=\int_{-4.5}^{-3.5}\int_{5}^{7.5}(\frac{(34e^{(- y - z)}ln(x))}{3})dydz|_{9}^{11.5}\\&\int_{-4.5}^{-3.5}\int_{5}^{7.5}(\frac{(34e^{(- y - z)}ln(\frac{23}{18}))}{3})dydz=\int_{-4.5}^{-3.5}(-\frac{(34e^{(- y - z)}ln(\frac{23}{18}))}{3})dz|_{5}^{7.5}\\&\int_{-4.5}^{-3.5}(\frac{(34e^{(- z -\frac{ 15}{2})}ln(\frac{23}{18})\cdot(e^{(\frac{5}{2})} - 1))}{3})dz=-\frac{(34e^{(- z -\frac{ 15}{2})}ln(\frac{23}{18})\cdot(e^{(\frac{5}{2})} - 1))}{3}|_{-4.5}^{-3.5}=0.98\end{align*}$

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Example Question #793 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{8.5}\int_{-6}^{-4.5}\int_{-9}^{-7}(\frac{(38x^{2}y^{2}z)}{3})dxdydz\end{align*}$

Possible Answers:

$1.23\cdot10^{6}$

$-4.918\cdot10^{6}$

$3.689\cdot10^{6}$

Correct answer:

$1.23\cdot10^{6}$

Explanation:

$\begin{align*}&\int_{6}^{8.5}\int_{-6}^{-4.5}\int_{-9}^{-7}(\frac{(38x^{2}y^{2}z)}{3})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{8.5}\int_{-6}^{-4.5}\int_{-9}^{-7}(\frac{(38x^{2}y^{2}z)}{3})dxdydz=\int_{6}^{8.5}\int_{-6}^{-4.5}(\frac{(38x^{3}y^{2}z)}{9})dydz|_{-9}^{-7}\\&\int_{6}^{8.5}\int_{-6}^{-4.5}(\frac{(14668y^{2}z)}{9})dydz=\int_{6}^{8.5}(\frac{(14668y^{3}z)}{27})dz|_{-6}^{-4.5}\\&\int_{6}^{8.5}(\frac{(135679z)}{2})dz=\frac{(135679z^{2})}{4}|_{6}^{8.5}=1.23\cdot10^{6}\end{align*}$

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Example Question #311 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{9.5}\int_{9}^{10}\int_{8}^{9}(\frac{(9cos(z + 1)sin(y + 2))}{49})dxdydz\end{align*}$

Possible Answers:

0.24

-0.06

-0.95

0.06

Correct answer:

0.24

Explanation:

$\begin{align*}&\int_{6}^{9.5}\int_{9}^{10}\int_{8}^{9}(\frac{(9cos(z + 1)sin(y + 2))}{49})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{9.5}\int_{9}^{10}\int_{8}^{9}(\frac{(9cos(z + 1)sin(y + 2))}{49})dxdydz=\int_{6}^{9.5}\int_{9}^{10}(\frac{(9xcos(z + 1)sin(y + 2))}{49})dydz|_{8}^{9}\\&\int_{6}^{9.5}\int_{9}^{10}(\frac{(9cos(z + 1)sin(y + 2))}{49})dydz=\int_{6}^{9.5}(-\frac{(9cos(y + 2)cos(z + 1))}{49})dz|_{9}^{10}\\&\int_{6}^{9.5}(\frac{(9cos(z + 1)\cdot(cos(11) - cos(12)))}{49})dz=sin(z + 1)\cdot(\frac{(9cos(11))}{49}-\frac{ (9cos(12))}{49})|_{6}^{9.5}=0.24\end{align*}$

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Example Question #795 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{10.5}\int_{-3.5}^{-1.5}\int_{10}^{15}(\frac{(19cos(z + 2)sin(4x))}{(6\cdot3^y)})dxdydz\end{align*}$

Possible Answers:

-23.95

7.98

-1.6

3.99

Correct answer:

7.98

Explanation:

$\begin{align*}&\int_{9}^{10.5}\int_{-3.5}^{-1.5}\int_{10}^{15}(\frac{(19cos(z + 2)sin(4x))}{(6\cdot3^y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{10.5}\int_{-3.5}^{-1.5}\int_{10}^{15}(\frac{(19cos(z + 2)sin(4x))}{(6\cdot3^y)})dxdydz=\int_{9}^{10.5}\int_{-3.5}^{-1.5}(-\frac{(19cos(4x)cos(z + 2))}{(24\cdot3^y)})dydz|_{10}^{15}\\&\int_{9}^{10.5}\int_{-3.5}^{-1.5}(\frac{(19cos(z + 2)\cdot(\frac{cos(40)}{4}-\frac{ cos(60)}{4}))}{(6\cdot3^y)})dydz=\int_{9}^{10.5}(-\frac{(19cos(z + 2)\cdot(cos(40) - cos(60)))}{(24\cdot3^yln(3))})dz|_{-3.5}^{-1.5}\\&\int_{9}^{10.5}(\frac{(19\cdot3^{(\frac{1}{2})}cos(z + 2)\cdot(cos(40) - cos(60)))}{ln(3)})dz=\frac{(19\cdot3^{(\frac{1}{2})}sin(z + 2)\cdot(cos(40) - cos(60)))}{ln(3)}|_{9}^{10.5}=7.98\end{align*}$

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Example Question #796 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{10.5}\int_{8}^{11.5}\int_{10}^{13}(\frac{(58z^{2}cos(x + 1)cos(3y))}{11})dxdydz\end{align*}$

Possible Answers:

305

-228.7

914.9

-914.9

Correct answer:

914.9

Explanation:

$\begin{align*}&\int_{7}^{10.5}\int_{8}^{11.5}\int_{10}^{13}(\frac{(58z^{2}cos(x + 1)cos(3y))}{11})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{10.5}\int_{8}^{11.5}\int_{10}^{13}(\frac{(58z^{2}cos(x + 1)cos(3y))}{11})dxdydz=\int_{7}^{10.5}\int_{8}^{11.5}(\frac{(58z^{2}cos(3y)sin(x + 1))}{11})dydz|_{10}^{13}\\&\int_{7}^{10.5}\int_{8}^{11.5}(-\frac{(58z^{2}cos(3y)\cdot(sin(11) - sin(14)))}{11})dydz=\int_{7}^{10.5}(-\frac{(58z^{2}sin(3y)\cdot(sin(11) - sin(14)))}{33})dz|_{8}^{11.5}\\&\int_{7}^{10.5}(\frac{(58z^{2}\cdot(sin(11) - sin(14))\cdot(sin(24) - sin(\frac{69}{2})))}{33})dz=z^{3}\cdot(\frac{(58sin(11)sin(24))}{99}-\frac{ (58sin(14)sin(24))}{99}-\frac{ (58sin(11)sin(\frac{69}{2}))}{99}+\frac{ (58sin(14)sin(\frac{69}{2}))}{99})|_{7}^{10.5}=914.9\end{align*}$

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Example Question #797 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{7.5}\int_{-4}^{0.5}\int_{-3.5}^{-2.5}(\frac{(7e^{(-2x)}e^{(-y)})}{(6z^{2})})dxdydz\end{align*}$

Possible Answers:

248.9

995.5

-165.9

Correct answer:

995.5

Explanation:

$\begin{align*}&\int_{6}^{7.5}\int_{-4}^{0.5}\int_{-3.5}^{-2.5}(\frac{(7e^{(-2x)}e^{(-y)})}{(6z^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{7.5}\int_{-4}^{0.5}\int_{-3.5}^{-2.5}(\frac{(7e^{(-2x)}e^{(-y)})}{(6z^{2})})dxdydz=\int_{6}^{7.5}\int_{-4}^{0.5}(-\frac{(7e^{(- 2x - y)})}{(12z^{2})})dydz|_{-3.5}^{-2.5}\\&\int_{6}^{7.5}\int_{-4}^{0.5}(\frac{(7e^{(5)}e^{(-y)}\cdot(e^{(2)} - 1))}{(12z^{2})})dydz=\int_{6}^{7.5}(-\frac{(7e^{(5 - y)}\cdot(e^{(2)} - 1))}{(12z^{2})})dz|_{-4}^{0.5}\\&\int_{6}^{7.5}(\frac{(7e^{(\frac{9}{2})}\cdot(e^{(2)} - 1)\cdot(e^{(\frac{9}{2})} - 1))}{(12z^{2})})dz=\frac{(\frac{(7e^{(9)})}{12}-\frac{ (7e^{(\frac{9}{2})})}{12}-\frac{ (7e^{(11)})}{12}+\frac{ (7e^{(\frac{13}{2})})}{12})}{z}|_{6}^{7.5}=995.5\end{align*}$

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Example Question #321 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-8}\int_{8}^{11.5}\int_{9}^{10}(\frac{(37z)}{(6x)})dxdydz\end{align*}$

Possible Answers:

-38.66

-19.33

9.66

57.99

Correct answer:

-19.33

Explanation:

$\begin{align*}&\int_{-9}^{-8}\int_{8}^{11.5}\int_{9}^{10}(\frac{(37z)}{(6x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-8}\int_{8}^{11.5}\int_{9}^{10}(\frac{(37z)}{(6x)})dxdydz=\int_{-9}^{-8}\int_{8}^{11.5}(\frac{(37zln(x))}{6})dydz|_{9}^{10}\\&\int_{-9}^{-8}\int_{8}^{11.5}(\frac{(37zln(\frac{10}{9}))}{6})dydz=\int_{-9}^{-8}(\frac{(37yzln(\frac{10}{9}))}{6})dz|_{8}^{11.5}\\&\int_{-9}^{-8}(\frac{(259zln(\frac{10}{9}))}{12})dz=\frac{(259z^{2}ln(\frac{10}{9}))}{24}|_{-9}^{-8}=-19.33\end{align*}$

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Example Question #322 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{5}^{8}\int_{-3.5}^{1}\int_{-8}^{-5}(\frac{e^{(2z)}}{(7\cdot2^yx^{2})})dxdydz\end{align*}$

Possible Answers:

$-1.482\cdot10^{6}$

246900

740800

Correct answer:

740800

Explanation:

$\begin{align*}&\int_{5}^{8}\int_{-3.5}^{1}\int_{-8}^{-5}(\frac{e^{(2z)}}{(7\cdot2^yx^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{5}^{8}\int_{-3.5}^{1}\int_{-8}^{-5}(\frac{e^{(2z)}}{(7\cdot2^yx^{2})})dxdydz=\int_{5}^{8}\int_{-3.5}^{1}(-\frac{e^{(2z)}}{(7\cdot2^yx)})dydz|_{-8}^{-5}\\&\int_{5}^{8}\int_{-3.5}^{1}(\frac{(3e^{(2z)})}{(280\cdot2^y)})dydz=\int_{5}^{8}(-\frac{(3e^{(2z)})}{(280\cdot2^yln(2))})dz|_{-3.5}^{1}\\&\int_{5}^{8}(\frac{(3e^{(2z)}\cdot(8\cdot2^{(\frac{1}{2})} -\frac{ 1}{2}))}{(280ln(2))})dz=\frac{(3e^{(2z)}\cdot(8\cdot2^{(\frac{1}{2})} -\frac{ 1}{2}))}{(560ln(2))}|_{5}^{8}=740800\end{align*}$

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Example Question #321 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{10.5}\int_{-8}^{-4.5}\int_{10}^{14}(\frac{(3ycos(4x)sin(z + 2))}{5})dxdydz\end{align*}$

Possible Answers:

-2.65

15.87

-7.93

3.97

Correct answer:

-7.93

Explanation:

$\begin{align*}&\int_{7}^{10.5}\int_{-8}^{-4.5}\int_{10}^{14}(\frac{(3ycos(4x)sin(z + 2))}{5})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{10.5}\int_{-8}^{-4.5}\int_{10}^{14}(\frac{(3ycos(4x)sin(z + 2))}{5})dxdydz=\int_{7}^{10.5}\int_{-8}^{-4.5}(\frac{(3ysin(4x)sin(z + 2))}{20})dydz|_{10}^{14}\\&\int_{7}^{10.5}\int_{-8}^{-4.5}(-\frac{(3ysin(z + 2)\cdot(sin(40) - sin(56)))}{20})dydz=\int_{7}^{10.5}(-\frac{(3y^{2}sin(z + 2)\cdot(sin(40) - sin(56)))}{40})dz|_{-8}^{-4.5}\\&\int_{7}^{10.5}(\frac{(105sin(z + 2)\cdot(sin(40) - sin(56)))}{32})dz=-cos(z + 2)\cdot(\frac{(105sin(40))}{32}-\frac{ (105sin(56))}{32})|_{7}^{10.5}=-7.93\end{align*}$

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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 #791 : Calculus 3

Example Question #311 : Multiple Integration

Example Question #793 : Calculus 3

Example Question #311 : Triple Integrals

Example Question #795 : Calculus 3

Example Question #796 : Calculus 3

Example Question #797 : Calculus 3

Example Question #321 : Multiple Integration

Example Question #322 : Multiple Integration

Example Question #321 : Triple Integrals

All Calculus 3 Resources

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