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Calculus 3 : Multiple Integration

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Example Questions

Example Question #331 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-4}\int_{2}^{3.5}\int_{3.5}^{5}(\frac{(9cos(4z)e^{(2x)})}{(73y)})dxdydz\end{align*}$

Possible Answers:

-111.5

55.75

669

Correct answer:

-111.5

Explanation:

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

$\begin{align*}&\int_{-6}^{-4}\int_{2}^{3.5}\int_{3.5}^{5}(\frac{(9cos(4z)e^{(2x)})}{(73y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-4}\int_{2}^{3.5}\int_{3.5}^{5}(\frac{(9cos(4z)e^{(2x)})}{(73y)})dxdydz=\int_{-6}^{-4}\int_{2}^{3.5}(\frac{(9cos(4z)e^{(2x)})}{(146y)})dydz|_{3.5}^{5}\\&\int_{-6}^{-4}\int_{2}^{3.5}(\frac{(9cos(4z)e^{(7)}\cdot(e^{(3)} - 1))}{(146y)})dydz=\int_{-6}^{-4}(-ln(y)\cdot(\frac{(9cos(4z)e^{(7)})}{146}-\frac{ (9cos(4z)e^{(10)})}{146}))dz|_{2}^{3.5}\\&\int_{-6}^{-4}(\frac{(9cos(4z)e^{(7)}ln(\frac{7}{4})\cdot(e^{(3)} - 1))}{146})dz=\frac{(9sin(4z)e^{(7)}ln(\frac{7}{4})\cdot(e^{(3)} - 1))}{584}|_{-6}^{-4}=-111.5\end{align*}$

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Example Question #199 : Triple Integration In Cartesian Coordinates

Evaluate: $\int_{0}^{1}\int_{0}^{2}\int_{0}^{3}x^3 \:dz \:dy\:dx$

Possible Answers:

$\frac{45}{4}$

$\frac{81}{2}$

$\frac{3}{2}$

$\frac{6}{5}$

Correct answer:

$\frac{3}{2}$

Explanation:

Evaluate $\int_{0}^{3}x^3 \:dz$ first.

$\int_{0}^{3}x^3 \:dz = x^3z|_{0}^{3} = 3x^3$

Substitute this back and evaluate $\int_{0}^{2}3x^3 \: dy$ .

$\int_{0}^{2}3x^3 \: dy = 3x^3y |_{0}^{2} = 6x^3$

Substitute this value and evaluate the last integral $\int_{0}^{1} 6x^3 dx$ .

$\int_{0}^{1} 6x^3 dx = \frac{6x^4}{4}|_{0}^{1}= \frac{3x^4}{2}|_{0}^{1} = \frac{3(1)^4}{2} = \frac{3}{2}$

The answer is: $\frac{3}{2}$

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Example Question #198 : Triple Integration In Cartesian Coordinates

Evaluate the following integral over the volume V which is enclosed by a cube of side length 2 in the first octant, with one corner on the origin.

$\int\int\int_V\frac{xyz}{8}\:dx\:dy\:dz$

Possible Answers:

$\frac{1}{8}$

$\frac{1}{64}$

Correct answer:

Explanation:

The description of the cube requires: 0<x<2, 0<y<2, and 0<z<2. The integral becomes:

$\int^2_0\int^2_0\int_0^2\frac{xyz}{8}\:dx\:dy\:dz$

To solve this integral:

$\int^2_0\int^2_0(\frac{x^2}{2}|^2_0)\frac{yz}{8}\:dy\:dz=\int^2_0\int^2_0\frac{yz}{4}\:dy\:dz$

$= \int^2_0(\frac{y^2}{2}|^2_0)\frac{z}{4}\:dz=\int^2_0\frac{z}{2}dz= \frac{z^2}{4}|^2_0=1$

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Example Question #201 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-4}^{-2}\int_{-10}^{-7}\int_{-6}^{-4.5}(\frac{(12cos(3y)sin(3x)e^{(-z)})}{79})dxdydz\end{align*}$

Possible Answers:

-0.1

0.29

0.03

-0.02

Correct answer:

-0.1

Explanation:

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

$\begin{align*}&\int_{-4}^{-2}\int_{-10}^{-7}\int_{-6}^{-4.5}(\frac{(12cos(3y)sin(3x)e^{(-z)})}{79})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4}^{-2}\int_{-10}^{-7}\int_{-6}^{-4.5}(\frac{(12cos(3y)sin(3x)e^{(-z)})}{79})dxdydz=\int_{-4}^{-2}\int_{-10}^{-7}(-\frac{(4cos(3x)cos(3y)e^{(-z)})}{79})dydz|_{-6}^{-4.5}\\&\int_{-4}^{-2}\int_{-10}^{-7}(\frac{(4cos(3y)e^{(-z)}\cdot (cos(18) - cos(\frac{27}{2})))}{79})dydz=\int_{-4}^{-2}(\frac{(4sin(3y)e^{(-z)}\cdot (cos(18) - cos(\frac{27}{2})))}{237})dz|_{-10}^{-7}\\&\int_{-4}^{-2}(-\frac{(4e^{(-z)}\cdot (cos(18) - cos(\frac{27}{2}))\cdot (sin(21) - sin(30)))}{237})dz=\frac{(4e^{(-z)}\cdot (cos(18) - cos(\frac{27}{2}))\cdot (sin(21) - sin(30)))}{237}|_{-4}^{-2}=-0.1\end{align*}$

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Example Question #201 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{3.5}^{8.5}\int_{8}^{13}\int_{4.5}^{7}(\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-x)})}{(8y)})dxdydz\end{align*}$

Possible Answers:

-0.09

0.36

-0.73

0.73

Correct answer:

0.36

Explanation:

$\begin{align*}&\int_{3.5}^{8.5}\int_{8}^{13}\int_{4.5}^{7}(\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-x)})}{(8y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3.5}^{8.5}\int_{8}^{13}\int_{4.5}^{7}(\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-x)})}{(8y)})dxdydz=\int_{3.5}^{8.5}\int_{8}^{13}(-\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-x)})}{(8y)})dydz|_{4.5}^{7}\\&\int_{3.5}^{8.5}\int_{8}^{13}(\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-7)}\cdot (e^{(\frac{5}{2})} - 1))}{(8y)})dydz=\int_{3.5}^{8.5}(\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-7)}ln(y)\cdot (e^{(\frac{5}{2})} - 1))}{8})dz|_{8}^{13}\\&\int_{3.5}^{8.5}(\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-7)}ln(\frac{13}{8})\cdot (e^{(\frac{5}{2})} - 1))}{8})dz=\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-7)}ln(\frac{13}{8})\cdot (e^{(\frac{5}{2})} - 1))}{(4ln(2))}|_{3.5}^{8.5}=0.36\end{align*}$

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Example Question #202 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-2.5}\int_{-5}^{2.5}\int_{6}^{7.5}(\frac{(3^{(\frac{y}{2})}cos(3z))}{(18x)})dxdydz\end{align*}$

Possible Answers:

0.02

0.1

-0.15

-0.05

Correct answer:

-0.05

Explanation:

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

$\begin{align*}&\int_{-6}^{-2.5}\int_{-5}^{2.5}\int_{6}^{7.5}(\frac{(3^{(\frac{y}{2})}cos(3z))}{(18x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-2.5}\int_{-5}^{2.5}\int_{6}^{7.5}(\frac{(3^{(\frac{y}{2})}cos(3z))}{(18x)})dxdydz=\int_{-6}^{-2.5}\int_{-5}^{2.5}(\frac{(3^{(\frac{y}{2})}cos(3z)ln(x))}{18})dydz|_{6}^{7.5}\\&\int_{-6}^{-2.5}\int_{-5}^{2.5}(\frac{(3^{(\frac{y}{2})}cos(3z)ln(\frac{5}{4}))}{18})dydz=\int_{-6}^{-2.5}(\frac{(3^{(\frac{y}{2})}cos(3z)ln(\frac{5}{4}))}{(9ln(3))})dz|_{-5}^{2.5}\\&\int_{-6}^{-2.5}(-\frac{(cos(3z)ln(\frac{5}{4})\cdot (2\cdot 3^{(\frac{1}{2})} - 162\cdot 3^{(\frac{1}{4})}))}{(486ln(3))})dz=-\frac{(sin(3z)ln(\frac{5}{4})\cdot (2\cdot 3^{(\frac{1}{2})} - 162\cdot 3^{(\frac{1}{4})}))}{(1458ln(3))}|_{-6}^{-2.5}=-0.05\end{align*}$

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Example Question #203 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{11}\int_{10}^{12.5}\int_{4.5}^{9}(\frac{(54\cdot 3^{(\frac{x}{3})}cos(3z)sin(3y))}{11})dxdydz\end{align*}$

Possible Answers:

-281.8

23.48

-46.97

187.9

Correct answer:

-46.97

Explanation:

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

$\begin{align*}&\int_{6}^{11}\int_{10}^{12.5}\int_{4.5}^{9}(\frac{(54\cdot 3^{(\frac{x}{3})}cos(3z)sin(3y))}{11})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{11}\int_{10}^{12.5}\int_{4.5}^{9}(\frac{(54\cdot 3^{(\frac{x}{3})}cos(3z)sin(3y))}{11})dxdydz=\int_{6}^{11}\int_{10}^{12.5}(\frac{(162\cdot 3^{(\frac{x}{3})}cos(3z)sin(3y))}{(11ln(3))})dydz|_{4.5}^{9}\\&\int_{6}^{11}\int_{10}^{12.5}(-\frac{(486cos(3z)sin(3y)\cdot (3^{(\frac{1}{2})} - 9))}{(11ln(3))})dydz=\int_{6}^{11}(\frac{(162cos(3y)cos(3z)\cdot (3^{(\frac{1}{2})} - 9))}{(11ln(3))})dz|_{10}^{12.5}\\&\int_{6}^{11}(-\frac{(162cos(3z)\cdot (cos(30) - cos(\frac{75}{2}))\cdot (3^{(\frac{1}{2})} - 9))}{(11ln(3))})dz=-\frac{(54sin(3z)\cdot (cos(30) - cos(\frac{75}{2}))\cdot (3^{(\frac{1}{2})} - 9))}{(11ln(3))}|_{6}^{11}=-46.97\end{align*}$

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Example Question #331 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-5.5}\int_{-10}^{-9}\int_{-4.5}^{0.5}(\frac{(5sin(y + 2)e^{(2x)})}{(6z^{2})})dxdydz\end{align*}$

Possible Answers:

0.02

0.07

-0.07

-0.22

Correct answer:

-0.07

Explanation:

$\begin{align*}&\int_{-9}^{-5.5}\int_{-10}^{-9}\int_{-4.5}^{0.5}(\frac{(5sin(y + 2)e^{(2x)})}{(6z^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-5.5}\int_{-10}^{-9}\int_{-4.5}^{0.5}(\frac{(5sin(y + 2)e^{(2x)})}{(6z^{2})})dxdydz=\int_{-9}^{-5.5}\int_{-10}^{-9}(\frac{(5sin(y + 2)e^{(2x)})}{(12z^{2})})dydz|_{-4.5}^{0.5}\\&\int_{-9}^{-5.5}\int_{-10}^{-9}(\frac{(5sin(y + 2)e^{(-9)}\cdot (e^{(10)} - 1))}{(12z^{2})})dydz=\int_{-9}^{-5.5}(-\frac{(5cos(y + 2)e^{(-9)}\cdot (e^{(10)} - 1))}{(12z^{2})})dz|_{-10}^{-9}\\&\int_{-9}^{-5.5}(-\frac{(5e^{(-9)}\cdot (cos(7) - cos(8))\cdot (e^{(10)} - 1))}{(12z^{2})})dz=\frac{(5e^{(-9)}\cdot (cos(7) - cos(8))\cdot (e^{(10)} - 1))}{(12z)}|_{-9}^{-5.5}=-0.07\end{align*}$

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Example Question #204 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-5}^{-1.5}\int_{6}^{9.5}\int_{-8}^{-5.5}(8\cdot 2^{(\frac{z}{2})}cos(x + 1)cos(4y))dxdydz\end{align*}$

Possible Answers:

-0.95

4.74

1.58

-9.47

Correct answer:

4.74

Explanation:

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

$\begin{align*}&\int_{-5}^{-1.5}\int_{6}^{9.5}\int_{-8}^{-5.5}(8\cdot 2^{(\frac{z}{2})}cos(x + 1)cos(4y))dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-5}^{-1.5}\int_{6}^{9.5}\int_{-8}^{-5.5}(8\cdot 2^{(\frac{z}{2})}cos(x + 1)cos(4y))dxdydz=\int_{-5}^{-1.5}\int_{6}^{9.5}(8\cdot 2^{(\frac{z}{2})}cos(4y)sin(x + 1))dydz|_{-8}^{-5.5}\\&\int_{-5}^{-1.5}\int_{6}^{9.5}(8\cdot 2^{(\frac{z}{2})}cos(4y)\cdot (sin(7) - sin(\frac{9}{2})))dydz=\int_{-5}^{-1.5}(2\cdot 2^{(\frac{z}{2})}sin(4y)\cdot (sin(7) - sin(\frac{9}{2})))dz|_{6}^{9.5}\\&\int_{-5}^{-1.5}(-2\cdot 2^{(\frac{z}{2})}\cdot (sin(7) - sin(\frac{9}{2}))\cdot (sin(24) - sin(38)))dz=-\frac{(4\cdot 2^{(\frac{z}{2})}\cdot (\frac{cos(17)}{2}- cos(31) -\frac{ cos(\frac{39}{2})}{2}+\frac{ cos(45)}{2}+\frac{ cos(\frac{57}{2})}{2}+\frac{ cos(\frac{67}{2})}{2}-\frac{ cos(\frac{85}{2})}{2}))}{ln(2)}|_{-5}^{-1.5}=4.74\end{align*}$

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Example Question #205 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{5.5}\int_{-3.5}^{1.5}\int_{5}^{7.5}(\frac{(e^{(2y)}e^{(z)})}{(5x)})dxdydz\end{align*}$

Possible Answers:

182.9

1097

-45.73

-182.9

Correct answer:

182.9

Explanation:

$\begin{align*}&\int_{3}^{5.5}\int_{-3.5}^{1.5}\int_{5}^{7.5}(\frac{(e^{(2y)}e^{(z)})}{(5x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{5.5}\int_{-3.5}^{1.5}\int_{5}^{7.5}(\frac{(e^{(2y)}e^{(z)})}{(5x)})dxdydz=\int_{3}^{5.5}\int_{-3.5}^{1.5}(\frac{(e^{(2y + z)}ln(x))}{5})dydz|_{5}^{7.5}\\&\int_{3}^{5.5}\int_{-3.5}^{1.5}(\frac{(e^{(2y + z)}ln(\frac{3}{2}))}{5})dydz=\int_{3}^{5.5}(\frac{(e^{(2y + z)}ln(\frac{3}{2}))}{10})dz|_{-3.5}^{1.5}\\&\int_{3}^{5.5}(\frac{(e^{(-7)}e^{(z)}ln(\frac{3}{2})\cdot (e^{(10)} - 1))}{10})dz=\frac{(e^{(z - 7)}ln(\frac{3}{2})\cdot (e^{(10)} - 1))}{10}|_{3}^{5.5}=182.9\end{align*}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #331 : Triple Integrals

Example Question #199 : Triple Integration In Cartesian Coordinates

Example Question #198 : Triple Integration In Cartesian Coordinates

Example Question #201 : Triple Integration In Cartesian Coordinates

Example Question #201 : Triple Integration In Cartesian Coordinates

Example Question #202 : Triple Integration In Cartesian Coordinates

Example Question #203 : Triple Integration In Cartesian Coordinates

Example Question #331 : Multiple Integration

Example Question #204 : Triple Integration In Cartesian Coordinates

Example Question #205 : Triple Integration In Cartesian Coordinates

All Calculus 3 Resources

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