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Calculus 3 : Multiple Integration

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Example Questions

Example Question #71 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{12}\int_{-7}^{-4}\int_{-10}^{-9}(e^{(3z)}\cdot(y +\frac{ 43}{10}))dxdydz\end{align*}$

Possible Answers:

$2.58\cdot10^{16}$

$-5.16\cdot10^{15}$

$1.29\cdot10^{15}$

$-2.58\cdot10^{16}$

Correct answer:

$-5.16\cdot10^{15}$

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{10}^{12}\int_{-7}^{-4}\int_{-10}^{-9}(e^{(3z)}\cdot(y +\frac{ 43}{10}))dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{12}\int_{-7}^{-4}\int_{-10}^{-9}(e^{(3z)}\cdot(y +\frac{ 43}{10}))dxdydz=\int_{10}^{12}\int_{-7}^{-4}(xe^{(3z)}\cdot(y +\frac{ 43}{10}))dydz|_{-10}^{-9}\\&\int_{10}^{12}\int_{-7}^{-4}(e^{(3z)}\cdot(y + 4.3))dydz=\int_{10}^{12}(0.5e^{(3z)}\cdot(y + 4.3)^{2})dz|_{-7}^{-4}\\&\int_{10}^{12}(-3.6e^{(3z)})dz=-1.2e^{(3z)}dz|_{10}^{12}=-5.1607\cdot10^{15}\end{align*}$

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Example Question #72 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-10}^{-9}\int_{6}^{11}\int_{9}^{10}(\frac{(49cos(4y)sin(4x))}{4})dxdydz\end{align*}$

Possible Answers:

0.76

0.06

-1.91

0.38

Correct answer:

0.38

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-10}^{-9}\int_{6}^{11}\int_{9}^{10}(\frac{(49cos(4y)sin(4x))}{4})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-10}^{-9}\int_{6}^{11}\int_{9}^{10}(\frac{(49cos(4y)sin(4x))}{4})dxdydz=\int_{-10}^{-9}\int_{6}^{11}(-3.062cos(4x)cos(4y))dydz|_{9}^{10}\\&\int_{-10}^{-9}\int_{6}^{11}(1.65cos(4y))dydz=\int_{-10}^{-9}(0.4127sin(4y))dz|_{6}^{11}\\&\int_{-10}^{-9}(0.381)dz=0.381z|_{-10}^{-9}=0.38\end{align*}$

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Example Question #73 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-8}^{-6}\int_{7}^{8}\int_{-10}^{-7}(z +\frac{ (37cos(4x)cos(y + 2))}{2})dxdydz\end{align*}$

Possible Answers:

11.55

138.6

-46.19

Correct answer:

-46.19

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-8}^{-6}\int_{7}^{8}\int_{-10}^{-7}(z +\frac{ (37cos(4x)cos(y + 2))}{2})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-8}^{-6}\int_{7}^{8}\int_{-10}^{-7}(z +\frac{ (37cos(4x)cos(y + 2))}{2})dxdydz=\int_{-8}^{-6}\int_{7}^{8}(xz + 4.625cos(y + 2)sin(4x))dydz|_{-10}^{-7}\\&\int_{-8}^{-6}\int_{7}^{8}(3z + 2.19cos(y + 2))dydz=\int_{-8}^{-6}(3yz + 4.386cos(0.5y + 1)sin(0.5y + 1))dz|_{7}^{8}\\&\int_{-8}^{-6}(3z - 2.1)dz=1.5z^{2} - 2.097z|_{-8}^{-6}=-46.19\end{align*}$

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Example Question #211 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-10}^{-6}\int_{9}^{11}\int_{-6}^{-1}(6sin(z + 1))dxdydz\end{align*}$

Possible Answers:

-71.69

430.1

-17.92

-215.1

Correct answer:

-71.69

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-10}^{-6}\int_{9}^{11}\int_{-6}^{-1}(6sin(z + 1))dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-10}^{-6}\int_{9}^{11}\int_{-6}^{-1}(6sin(z + 1))dxdydz=\int_{-10}^{-6}\int_{9}^{11}(6xsin(z + 1))dydz|_{-6}^{-1}\\&\int_{-10}^{-6}\int_{9}^{11}(30sin(z + 1))dydz=\int_{-10}^{-6}(30ysin(z + 1))dz|_{9}^{11}\\&\int_{-10}^{-6}(60sin(z + 1))dz=-60cos(z + 1)|_{-10}^{-6}=-71.69\end{align*}$

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Example Question #212 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{2}^{5}\int_{-6}^{-1}\int_{9}^{14}(\frac{(sin(3y)e^{(4x)})}{(6z)})dxdydz\end{align*}$

Possible Answers:

$2.196\cdot10^{23}$

$4.393\cdot10^{22}$

$-8.786\cdot10^{21}$

$-2.636\cdot10^{23}$

Correct answer:

$4.393\cdot10^{22}$

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{5}\int_{-6}^{-1}\int_{9}^{14}(\frac{(sin(3y)e^{(4x)})}{(6z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{5}\int_{-6}^{-1}\int_{9}^{14}(\frac{(sin(3y)e^{(4x)})}{(6z)})dxdydz=\int_{2}^{5}\int_{-6}^{-1}(\frac{(0.04167e^{(4x)}sin(3y))}{z})dydz|_{9}^{14}\\&\int_{2}^{5}\int_{-6}^{-1}(\frac{(8.72\cdot10^{22}sin(3y))}{z})dydz=\int_{2}^{5}(-\frac{(2.905\cdot10^{22}cos(3y))}{z})dz|_{-6}^{-1}\\&\int_{2}^{5}(\frac{4.79\cdot10^{22}}{z})dz=4.794\cdot10^{22}ln(z)|_{2}^{5}=4.393\cdot10^{22}\end{align*}$

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Example Question #213 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-3}\int_{8}^{10}\int_{7}^{11}(\frac{(3\cdot2^{(4x)})}{(5y^{3}z^{2})})dxdydz\end{align*}$

Possible Answers:

$-1.02\cdot10^{10}$

$-4.079\cdot10^{8}$

$6.118\cdot10^{9}$

$2.0394\cdot10^{9}$

Correct answer:

$2.0394\cdot10^{9}$

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-7}^{-3}\int_{8}^{10}\int_{7}^{11}(\frac{(3\cdot2^{(4x)})}{(5y^{3}z^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-3}\int_{8}^{10}\int_{7}^{11}(\frac{(3\cdot2^{(4x)})}{(5y^{3}z^{2})})dxdydz=\int_{-7}^{-3}\int_{8}^{10}(\frac{(0.2164\cdot2^{(4x)})}{(y^{3}z^{2})})dydz|_{7}^{11}\\&\int_{-7}^{-3}\int_{8}^{10}(\frac{3.81\cdot10^{12}}{(y^{3}z^{2})})dydz=\int_{-7}^{-3}(-\frac{1.903\cdot10^{12}}{(y^{2}z^{2})})dz|_{8}^{10}\\&\int_{-7}^{-3}(\frac{1.07\cdot10^{10}}{z^{2}})dz=-\frac{1.071\cdot10^{10}}{z}|_{-7}^{-3}=2.0394\cdot10^{9}\end{align*}$

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Example Question #214 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{6}\int_{4}^{9}\int_{-8}^{-3}(\frac{(11x^{2}yz^{2})}{19})dxdydz\end{align*}$

Possible Answers:

739200

246399

$-1.232\cdot10^{6}$

Correct answer:

246399

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-3}^{6}\int_{4}^{9}\int_{-8}^{-3}(\frac{(11x^{2}yz^{2})}{19})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{6}\int_{4}^{9}\int_{-8}^{-3}(\frac{(11x^{2}yz^{2})}{19})dxdydz=\int_{-3}^{6}\int_{4}^{9}(0.193x^{3}yz^{2})dydz|_{-8}^{-3}\\&\int_{-3}^{6}\int_{4}^{9}(93.6yz^{2})dydz=\int_{-3}^{6}(46.8y^{2}z^{2})dz|_{4}^{9}\\&\int_{-3}^{6}(3044z^{2})dz=1014z^{3}|_{-3}^{6}=246399\end{align*}$

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Example Question #215 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-4}^{2}\int_{-6}^{-5}\int_{-9}^{-8}(\frac{(10\cdot3^ze^{(-2y)})}{(67\cdot3^x)})dxdydz\end{align*}$

Possible Answers:

$-2.052\cdot10^{8}$

$-2.052\cdot10^{9}$

$6.157\cdot10^{9}$

$1.026\cdot10^{9}$

Correct answer:

$1.026\cdot10^{9}$

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-4}^{2}\int_{-6}^{-5}\int_{-9}^{-8}(\frac{(10\cdot3^ze^{(-2y)})}{(67\cdot3^x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4}^{2}\int_{-6}^{-5}\int_{-9}^{-8}(\frac{(10\cdot3^ze^{(-2y)})}{(67\cdot3^x)})dxdydz=\int_{-4}^{2}\int_{-6}^{-5}(-0.1359\cdot3^{(z - x)}e^{}(-2y))dydz|_{-9}^{-8}\\&\int_{-4}^{2}\int_{-6}^{-5}(1788\cdot3^ze^{(-2y)})dydz=\int_{-4}^{2}(-891.4\cdot3^ze^{(-2y)})dz|_{-6}^{-5}\\&\int_{-4}^{2}(1.25\cdot10^{8}\cdot3^z)dz=1.142\cdot10^{8}\cdot3^z|_{-4}^{2}=1.0262\cdot10^{9}\end{align*}$

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Example Question #216 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-5}\int_{-7}^{-6}\int_{7}^{11}(sin(z + 2)\cdot(4y +\frac{ 34}{(7\cdot4^x)}))dxdydz\end{align*}$

Possible Answers:

-209.9

174.9

17.49

-34.98

Correct answer:

-34.98

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-6}^{-5}\int_{-7}^{-6}\int_{7}^{11}(sin(z + 2)\cdot(4y +\frac{ 34}{(7\cdot4^x)}))dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-5}\int_{-7}^{-6}\int_{7}^{11}(sin(z + 2)\cdot(4y +\frac{ 34}{(7\cdot4^x)}))dxdydz=\int_{-6}^{-5}\int_{-7}^{-6}(-0.2061sin(z + 2)\cdot(\frac{17.0}{2^{(2x)}}- 19.41xy))dydz|_{7}^{11}\\&\int_{-6}^{-5}\int_{-7}^{-6}(4.91\cdot10^{-8}sin(z + 2)\cdot(3.26\cdot10^{8}y + 4344))dydz=\int_{-6}^{-5}(7.545\cdot10^{-17}sin(z + 2)\cdot(3.256\cdot10^{8}y + 4335)^{2})dz|_{-7}^{-6}\\&\int_{-6}^{-5}(-104sin(z + 2))dz=104cos(z + 2)|_{-6}^{-5}=-34.98\end{align*}$

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Example Question #217 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-5}\int_{7}^{8}\int_{9}^{11}(3z + sin(3y)\cdot(3x +\frac{ 37}{11}))dxdydz\end{align*}$

Possible Answers:

1272

-254.5

50.89

-763.4

Correct answer:

-254.5

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-9}^{-5}\int_{7}^{8}\int_{9}^{11}(3z + sin(3y)\cdot(3x +\frac{ 37}{11}))dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-5}\int_{7}^{8}\int_{9}^{11}(3z + sin(3y)\cdot(3x +\frac{ 37}{11}))dxdydz=\int_{-9}^{-5}\int_{7}^{8}(3xz + 0.1667sin(3y)\cdot(3x + 3.364)^{2})dydz|_{9}^{11}\\&\int_{-9}^{-5}\int_{7}^{8}(6z + 66.7sin(3y))dydz=\int_{-9}^{-5}(6yz - 44.48cos(1.5y)^{2})dz|_{7}^{8}\\&\int_{-9}^{-5}(6z - 21.6)dz=3z^{2} - 21.62z|_{-9}^{-5}=-254.5\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #71 : Triple Integration In Cartesian Coordinates

Example Question #72 : Triple Integration In Cartesian Coordinates

Example Question #73 : Triple Integration In Cartesian Coordinates

Example Question #211 : Triple Integrals

Example Question #212 : Triple Integrals

Example Question #213 : Triple Integrals

Example Question #214 : Triple Integrals

Example Question #215 : Triple Integrals

Example Question #216 : Triple Integrals

Example Question #217 : Triple Integrals

All Calculus 3 Resources

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