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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #211 : Multiple Integration

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

Possible Answers:

-71.69

-215.1

430.1

-17.92

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 : Multiple Integration

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

Possible Answers:

$-2.636\cdot10^{23}$

$2.196\cdot10^{23}$

$4.393\cdot10^{22}$

$-8.786\cdot10^{21}$

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 : Multiple Integration

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

Possible Answers:

$-4.079\cdot10^{8}$

$2.0394\cdot10^{9}$

$6.118\cdot10^{9}$

$-1.02\cdot10^{10}$

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 : Multiple Integration

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

Possible Answers:

246399

739200

$-1.232\cdot10^{6}$

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 : Multiple Integration

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

Possible Answers:

$-2.052\cdot10^{9}$

$6.157\cdot10^{9}$

$-2.052\cdot10^{8}$

$1.026\cdot10^{9}$

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 : Multiple Integration

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

Possible Answers:

-34.98

17.49

-209.9

174.9

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 : Multiple Integration

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

Possible Answers:

-254.5

1272

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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Example Question #221 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-4}^{-2}\int_{-8}^{-7}\int_{-7}^{-3}(\frac{(49e^{(-y)})}{(5x^{3})})dxdydz\end{align*}$

Possible Answers:

-558.3

8375

Correct answer:

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-4}^{-2}\int_{-8}^{-7}\int_{-7}^{-3}(\frac{(49e^{(-y)})}{(5x^{3})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4}^{-2}\int_{-8}^{-7}\int_{-7}^{-3}(\frac{(49e^{(-y)})}{(5x^{3})})dxdydz=\int_{-4}^{-2}\int_{-8}^{-7}(-\frac{(4.9e^{(-y)})}{x^{2}})dydz|_{-7}^{-3}\\&\int_{-4}^{-2}\int_{-8}^{-7}(-0.444e^{(-y)})dydz=\int_{-4}^{-2}(0.4444e^{(-y)})dz|_{-8}^{-7}\\&\int_{-4}^{-2}(-837.0)dz=-837.5z|_{-4}^{-2}=-1675\end{align*}$

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Example Question #222 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-3}\int_{-9}^{-4}\int_{8}^{10}(\frac{(8\cdot3^{(4x)}\cdot3^{(2y)}sin(z + 1))}{29})dxdydz\end{align*}$

Possible Answers:

$-2.914\cdot10^{14}$

$3.643\cdot10^{14}$

$1.457\cdot10^{13}$

$7.285\cdot10^{13}$

Correct answer:

$7.285\cdot10^{13}$

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-7}^{-3}\int_{-9}^{-4}\int_{8}^{10}(\frac{(8\cdot3^{(4x)}\cdot3^{(2y)}sin(z + 1))}{29})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-3}\int_{-9}^{-4}\int_{8}^{10}(\frac{(8\cdot3^{(4x)}\cdot3^{(2y)}sin(z + 1))}{29})dxdydz=\int_{-7}^{-3}\int_{-9}^{-4}(0.06278\cdot3^{(4x)}\cdot3^{(2y)}sin(z + 1))dydz|_{8}^{10}\\&\int_{-7}^{-3}\int_{-9}^{-4}(7.63\cdot10^{17}\cdot3^{(2y)}sin(z + 1))dydz=\int_{-7}^{-3}(3.473\cdot10^{17}\cdot3^{(2y)}sin(z + 1))dz|_{-9}^{-4}\\&\int_{-7}^{-3}(5.29\cdot10^{13}sin(z + 1))dz=-5.293\cdot10^{13}cos(z + 1)|_{-7}^{-3}=7.285\cdot10^{13}\end{align*}$

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Example Question #223 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{13}\int_{6}^{7}\int_{7}^{11}(\frac{(3\cdot3^z)}{(80xy^{2})})dxdydz\end{align*}$

Possible Answers:

188

564

3384

Correct answer:

564

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{10}^{13}\int_{6}^{7}\int_{7}^{11}(\frac{(3\cdot3^z)}{(80xy^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{13}\int_{6}^{7}\int_{7}^{11}(\frac{(3\cdot3^z)}{(80xy^{2})})dxdydz=\int_{10}^{13}\int_{6}^{7}(\frac{(0.0375\cdot3^zln(x))}{y^{2}})dydz|_{7}^{11}\\&\int_{10}^{13}\int_{6}^{7}(\frac{(0.0169\cdot3^z)}{y^{2}})dydz=\int_{10}^{13}(-\frac{(0.0169\cdot3^z)}{y})dz|_{6}^{7}\\&\int_{10}^{13}(4.04\cdot10^{-4}\cdot3^z)dz=(4.04\cdot10^{-4}\cdot3^zln(3))|_{10}^{13}=564\end{align*}$

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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 #211 : Multiple Integration

Example Question #212 : Multiple Integration

Example Question #213 : Multiple Integration

Example Question #214 : Multiple Integration

Example Question #215 : Multiple Integration

Example Question #216 : Multiple Integration

Example Question #217 : Multiple Integration

Example Question #221 : Multiple Integration

Example Question #222 : Multiple Integration

Example Question #223 : Multiple Integration

All Calculus 3 Resources

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