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Calculus 3 : Multiple Integration

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Example Questions

Example Question #221 : Triple Integrals

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

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 #81 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

$3.643\cdot10^{14}$

$-2.914\cdot10^{14}$

$7.285\cdot10^{13}$

$1.457\cdot10^{13}$

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 : Triple Integrals

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

Possible Answers:

188

3384

564

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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Example Question #221 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-2}^{3}\int_{7}^{11}\int_{-10}^{-6}(\frac{(43cos(x + 1)cos(4z)sin(y + 2))}{2})dxdydz\end{align*}$

Possible Answers:

-18.2

12.14

-6.07

1.21

Correct answer:

-6.07

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-2}^{3}\int_{7}^{11}\int_{-10}^{-6}(\frac{(43cos(x + 1)cos(4z)sin(y + 2))}{2})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-2}^{3}\int_{7}^{11}\int_{-10}^{-6}(\frac{(43cos(x + 1)cos(4z)sin(y + 2))}{2})dxdydz=\int_{-2}^{3}\int_{7}^{11}(21.5cos(4z)sin(x + 1)sin(y + 2))dydz|_{-10}^{-6}\\&\int_{-2}^{3}\int_{7}^{11}(29.48cos(4z)sin(y + 2))dydz=\int_{-2}^{3}(-29.48cos(y + 2)cos(4z))dz|_{7}^{11}\\&\int_{-2}^{3}(-53.61cos(4z))dz=-13.4sin(4z)|_{-2}^{3}=-6.07\end{align*}$

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Example Question #221 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{11}\int_{-10}^{-9}\int_{6}^{10}(\frac{(9e^{(\frac{z}{3})})}{(49y)})dxdydz\end{align*}$

Possible Answers:

13.38

-3.35

-33.45

-6.69

Correct answer:

-6.69

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{7}^{11}\int_{-10}^{-9}\int_{6}^{10}(\frac{(9e^{(\frac{z}{3})})}{(49y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{11}\int_{-10}^{-9}\int_{6}^{10}(\frac{(9e^{(\frac{z}{3})})}{(49y)})dxdydz=\int_{7}^{11}\int_{-10}^{-9}(\frac{(9xe^{(\frac{z}{3})})}{(49y)})dydz|_{6}^{10}\\&\int_{7}^{11}\int_{-10}^{-9}(\frac{(36e^{(\frac{z}{3})})}{(49y)})dydz=\int_{7}^{11}(\frac{(36e^{(\frac{z}{3})}ln(y))}{49})dz|_{-10}^{-9}\\&\int_{7}^{11}(\frac{(36e^{(\frac{z}{3})}ln(\frac{9}{10}))}{49})dz=\frac{(108e^{(\frac{z}{3})}ln(\frac{9}{10}))}{49}|_{7}^{11}=-6.69\end{align*}$

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Example Question #222 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{5}\int_{-10}^{-8}\int_{7}^{12}(\frac{(27cos(4y)sin(z + 1))}{(8x^{2})})dxdydz\end{align*}$

Possible Answers:

-0.02

-0.05

0.08

0.01

Correct answer:

-0.02

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{3}^{5}\int_{-10}^{-8}\int_{7}^{12}(\frac{(27cos(4y)sin(z + 1))}{(8x^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{5}\int_{-10}^{-8}\int_{7}^{12}(\frac{(27cos(4y)sin(z + 1))}{(8x^{2})})dxdydz=\int_{3}^{5}\int_{-10}^{-8}(-\frac{(27cos(4y)sin(z + 1))}{(8x)})dydz|_{7}^{12}\\&\int_{3}^{5}\int_{-10}^{-8}(\frac{(45cos(4y)sin(z + 1))}{224})dydz=\int_{3}^{5}(\frac{(45sin(4y)sin(z + 1))}{896})dz|_{-10}^{-8}\\&\int_{3}^{5}(-\frac{(45sin(z + 1)\cdot(\frac{sin(32)}{4}-\frac{ sin(40)}{4}))}{224})dz=cos(z + 1)\cdot(\frac{(45sin(32))}{896}-\frac{ (45sin(40))}{896})|_{3}^{5}=-0.02\end{align*}$

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Example Question #221 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{12}\int_{-10}^{-9}\int_{-9}^{-6}(15\cdot2^ze^{(-x)})dxdydz\end{align*}$

Possible Answers:

$1.024\cdot10^{8}$

$5.119\cdot10^{8}$

$-2.047\cdot10^{9}$

$3.071\cdot10^{9}$

Correct answer:

$5.119\cdot10^{8}$

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{10}^{12}\int_{-10}^{-9}\int_{-9}^{-6}(15\cdot2^ze^{(-x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{12}\int_{-10}^{-9}\int_{-9}^{-6}(15\cdot2^ze^{(-x)})dxdydz=\int_{10}^{12}\int_{-10}^{-9}(-15\cdot2^ze^{(-x)})dydz|_{-9}^{-6}\\&\int_{10}^{12}\int_{-10}^{-9}(-15\cdot2^z\cdot(e^{(6)} - e^{(9)}))dydz=\int_{10}^{12}(-15\cdot2^zy\cdot(e^{(6)} - e^{(9)}))dz|_{-10}^{-9}\\&\int_{10}^{12}(-15\cdot2^z\cdot(e^{(6)} - e^{(9)}))dz=-\frac{(15\cdot2^z\cdot(e^{(6)} - e^{(9)}))}{ln(2)}|_{10}^{12}=5.119\cdot10^{8}\end{align*}$

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Example Question #223 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{3}\int_{-9}^{-8}\int_{-7}^{-3}(\frac{(8ysin(z + 1))}{(105x)})dxdydz\end{align*}$

Possible Answers:

-0.18

2.72

-4.54

0.91

Correct answer:

0.91

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{3}\int_{-9}^{-8}\int_{-7}^{-3}(\frac{(8ysin(z + 1))}{(105x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{3}\int_{-9}^{-8}\int_{-7}^{-3}(\frac{(8ysin(z + 1))}{(105x)})dxdydz=\int_{-1}^{3}\int_{-9}^{-8}(\frac{(8ysin(z + 1)ln(x))}{105})dydz|_{-7}^{-3}\\&\int_{-1}^{3}\int_{-9}^{-8}(\frac{(8ysin(z + 1)ln(\frac{3}{7}))}{105})dydz=\int_{-1}^{3}(\frac{(4y^{2}sin(z + 1)ln(\frac{3}{7}))}{105})dz|_{-9}^{-8}\\&\int_{-1}^{3}(-\frac{(68sin(z + 1)ln(\frac{3}{7}))}{105})dz=\frac{(68cos(z + 1)ln(\frac{3}{7}))}{105}|_{-1}^{3}=0.91\end{align*}$

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Example Question #91 : Triple Integration In Cartesian Coordinates

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

Possible Answers:

-0.04

-0.01

0.1

0.05

Correct answer:

-0.01

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{5}\int_{-6}^{-4}\int_{-9}^{-5}(\frac{(cos(3y)e^{(\frac{x}{4})})}{(5z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{5}\int_{-6}^{-4}\int_{-9}^{-5}(\frac{(cos(3y)e^{(\frac{x}{4})})}{(5z)})dxdydz=\int_{2}^{5}\int_{-6}^{-4}(\frac{(4cos(3y)e^{(\frac{x}{4})})}{(5z)})dydz|_{-9}^{-5}\\&\int_{2}^{5}\int_{-6}^{-4}(\frac{(4cos(3y)e^{(-\frac{9}{4})}\cdot(e^{(1)} - 1))}{(5z)})dydz=\int_{2}^{5}(\frac{(4sin(3y)e^{(-\frac{9}{4})}\cdot(e^{(1)} - 1))}{(15z)})dz|_{-6}^{-4}\\&\int_{2}^{5}(-\frac{(4e^{(-\frac{9}{4})}\cdot(sin(12) - sin(18))\cdot(e^{(1)} - 1))}{(15z)})dz=\frac{(e^{(-\frac{9}{4})}ln(z)\cdot(4sin(12) - 4sin(18) - 4e^{(1)}sin(12) + 4e^{(1)}sin(18)))}{15}|_{2}^{5}=-0.01\end{align*}$

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Example Question #224 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-8}^{-5}\int_{-9}^{-6}\int_{8}^{11}(\frac{(13cos(x + 2)sin(4y)sin(z + 2))}{10})dxdydz\end{align*}$

Possible Answers:

0.17

-0.34

1.35

-1.35

Correct answer:

-0.34

Explanation:

$\begin{align*}&\text{Performing a triple integral, the order of integration does}\\&\text{not entirely matter.}\\&\text{For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-8}^{-5}\int_{-9}^{-6}\int_{8}^{11}(\frac{(13cos(x + 2)sin(4y)sin(z + 2))}{10})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-8}^{-5}\int_{-9}^{-6}\int_{8}^{11}(\frac{(13cos(x + 2)sin(4y)sin(z + 2))}{10})dxdydz=\int_{-8}^{-5}\int_{-9}^{-6}(\frac{(13sin(x + 2)sin(4y)sin(z + 2))}{10})dydz|_{8}^{11}\\&\int_{-8}^{-5}\int_{-9}^{-6}(-\frac{(13sin(4y)sin(z + 2)\cdot(sin(10) - sin(13)))}{10})dydz=\int_{-8}^{-5}(\frac{(13cos(4y)sin(z + 2)\cdot(sin(10) - sin(13)))}{40})dz|_{-9}^{-6}\\&\int_{-8}^{-5}(\frac{(13sin(z + 2)\cdot(cos(24) - cos(36))\cdot(sin(10) - sin(13)))}{40})dz=-\frac{(13cos(z + 2)\cdot(cos(24) - cos(36))\cdot(sin(10) - sin(13)))}{40}|_{-8}^{-5}=-0.34\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #221 : Triple Integrals

Example Question #81 : Triple Integration In Cartesian Coordinates

Example Question #223 : Triple Integrals

Example Question #221 : Triple Integrals

Example Question #221 : Triple Integrals

Example Question #222 : Triple Integrals

Example Question #221 : Multiple Integration

Example Question #223 : Triple Integrals

Example Question #91 : Triple Integration In Cartesian Coordinates

Example Question #224 : Triple Integrals

All Calculus 3 Resources

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