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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #101 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

-0.22

3.33

-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 #711 : Calculus 3

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

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 #235 : Triple Integrals

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

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 #236 : Triple Integrals

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

Possible Answers:

0.09

-0.28

-0.83

1.38

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 #237 : Triple Integrals

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

Possible Answers:

-6.11

1.02

3.05

-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 #238 : Triple Integrals

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

Possible Answers:

-1.47

-8.81

26.44

4.41

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 #239 : Triple Integrals

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

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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Example Question #241 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{1}^{6}\int_{6}^{8}\int_{10}^{12}(\frac{(2\cdot2^z\cdot4^ycos(x + 2))}{25})dxdydz\end{align*}$

Possible Answers:

$1.937\cdot10^{6}$

$-2.422\cdot10^{6}$

484300

Correct answer:

484300

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{1}^{6}\int_{6}^{8}\int_{10}^{12}(\frac{(2\cdot2^z\cdot4^ycos(x + 2))}{25})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{1}^{6}\int_{6}^{8}\int_{10}^{12}(\frac{(2\cdot2^z\cdot4^ycos(x + 2))}{25})dxdydz=\int_{1}^{6}\int_{6}^{8}(\frac{(2\cdot2^{(2y + z)}sin(x + 2))}{25})dydz|_{10}^{12}\\&\int_{1}^{6}\int_{6}^{8}(-2^{(2y + z)}\cdot(\frac{(2sin(12))}{25}-\frac{ (2sin(14))}{25}))dydz=\int_{1}^{6}(-\frac{(2^{(2y + z)}\cdot(sin(12) - sin(14)))}{(25ln(2))})dz|_{6}^{8}\\&\int_{1}^{6}(-\frac{(12288\cdot2^z\cdot(sin(12) - sin(14)))}{(5ln(2))})dz=-\frac{(12288\cdot2^z\cdot(sin(12) - sin(14)))}{(5ln(2)^{2})}|_{1}^{6}=484300\end{align*}$

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Example Question #102 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-4}\int_{6}^{8}\int_{9}^{14}(\frac{(3\cdot4^{(\frac{z}{2})}cos(x + 2))}{(2y)})dxdydz\end{align*}$

Possible Answers:

0.02

0.15

-0.01

-0.15

Correct answer:

0.02

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-7}^{-4}\int_{6}^{8}\int_{9}^{14}(\frac{(3\cdot4^{(\frac{z}{2})}cos(x + 2))}{(2y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-4}\int_{6}^{8}\int_{9}^{14}(\frac{(3\cdot4^{(\frac{z}{2})}cos(x + 2))}{(2y)})dxdydz=\int_{-7}^{-4}\int_{6}^{8}(\frac{(3\cdot2^zsin(x + 2))}{(2y)})dydz|_{9}^{14}\\&\int_{-7}^{-4}\int_{6}^{8}(-\frac{(2^z\cdot(\frac{(3sin(11))}{2}-\frac{ (3sin(16))}{2}))}{y})dydz=\int_{-7}^{-4}(-\frac{(3\cdot2^zln(y)\cdot(sin(11) - sin(16)))}{2})dz|_{6}^{8}\\&\int_{-7}^{-4}(-\frac{(3\cdot2^zln(\frac{4}{3})\cdot(sin(11) - sin(16)))}{2})dz=-\frac{(3\cdot2^z\cdot(ln(\frac{4}{3})sin(11) - ln(\frac{4}{3})sin(16)))}{(2ln(2))}|_{-7}^{-4}=0.02\end{align*}$

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Example Question #110 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{8}\int_{-8}^{-3}\int_{6}^{9}(\frac{(17e^{(\frac{x}{3})})}{(6\cdot2^y\cdot4^{(\frac{z}{2})})})dxdydz\end{align*}$

Possible Answers:

3264

9792

-652.8

Correct answer:

3264

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{8}\int_{-8}^{-3}\int_{6}^{9}(\frac{(17e^{(\frac{x}{3})})}{(6\cdot2^y\cdot4^{(\frac{z}{2})})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{8}\int_{-8}^{-3}\int_{6}^{9}(\frac{(17e^{(\frac{x}{3})})}{(6\cdot2^y\cdot4^{(\frac{z}{2})})})dxdydz=\int_{4}^{8}\int_{-8}^{-3}(\frac{(17\cdot(\frac{1}{2})^{(y + z)}e^{(\frac{x}{3})})}{2})dydz|_{6}^{9}\\&\int_{4}^{8}\int_{-8}^{-3}(\frac{(17\cdot(\frac{1}{2})^{(y + z)}e^{(2)}\cdot(e^{(1)} - 1))}{2})dydz=\int_{4}^{8}(\frac{(\frac{17}{2^{(y}+ z)}\cdot(e^{(2)} - e^{(3)}))}{(2ln(2))})dz|_{-8}^{-3}\\&\int_{4}^{8}(\frac{(2108e^{(2)}\cdot(e^{(1)} - 1))}{(2^zln(2))})dz=\frac{(2108\cdot(e^{(2)} - e^{(3)}))}{(2^zln(2)^{2})}|_{4}^{8}=3264\end{align*}$

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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 #101 : Triple Integration In Cartesian Coordinates

Example Question #711 : Calculus 3

Example Question #235 : Triple Integrals

Example Question #236 : Triple Integrals

Example Question #237 : Triple Integrals

Example Question #238 : Triple Integrals

Example Question #239 : Triple Integrals

Example Question #241 : Triple Integrals

Example Question #102 : Triple Integration In Cartesian Coordinates

Example Question #110 : Triple Integration In Cartesian Coordinates

All Calculus 3 Resources

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