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Calculus 3 : Multiple Integration

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Example Questions

Example Question #471 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{11}\int_{-3.5}^{0.5}\int_{-8}^{-3.5}(\frac{(5\cdot 2^{(\frac{y}{2})}cos(3x))}{(8z^{2})})dxdydz\end{align*}$

Possible Answers:

$-2.63\cdot10^{-4}$

$1.05\cdot10^{-3}$

$1.75\cdot10^{-4}$

$-1.05\cdot10^{-3}$

Correct answer:

$-1.05\cdot10^{-3}$

Explanation:

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

$\begin{align*}&\int_{6}^{11}\int_{-3.5}^{0.5}\int_{-8}^{-3.5}(\frac{(5\cdot 2^{(\frac{y}{2})}cos(3x))}{(8z^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{11}\int_{-3.5}^{0.5}\int_{-8}^{-3.5}(\frac{(5\cdot 2^{(\frac{y}{2})}cos(3x))}{(8z^{2})})dxdydz=\int_{6}^{11}\int_{-3.5}^{0.5}(\frac{(5\cdot 2^{(\frac{y}{2})}sin(3x))}{(24z^{2})})dydz|_{-8}^{-3.5}\\&\int_{6}^{11}\int_{-3.5}^{0.5}(-\frac{(5\cdot 2^{(\frac{y}{2})}\cdot (\frac{sin(\frac{21}{2})}{3}-\frac{ sin(24)}{3}))}{(8z^{2})})dydz=\int_{6}^{11}(-\frac{(5\cdot 2^{(\frac{y}{2})}\cdot (sin(\frac{21}{2}) - sin(24)))}{(12z^{2}ln(2))})dz|_{-3.5}^{0.5}\\&\int_{6}^{11}(-\frac{(5\cdot 2^{(\frac{1}{4})}sin(\frac{21}{2}) - 5\cdot 2^{(\frac{1}{4})}sin(24))}{(16z^{2}ln(2))})dz=\frac{(2^{(\frac{1}{4})}\cdot (\frac{(5sin(\frac{21}{2}))}{16}-\frac{ (5sin(24))}{16}))}{(zln(2))}|_{6}^{11}=-1.05\cdot10^{-3}\end{align*}$

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Example Question #471 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{8}\int_{-9}^{-4}\int_{-10}^{-7}(\frac{1}{(x^{2}y^{2}z)})dxdydz\end{align*}$

Possible Answers:

$-8.56\cdot10^{-4}$

$1.03\cdot10^{-2}$

$1.71\cdot10^{-3}$

$-8.56\cdot10^{-3}$

Correct answer:

$1.71\cdot10^{-3}$

Explanation:

$\begin{align*}&\int_{6}^{8}\int_{-9}^{-4}\int_{-10}^{-7}(\frac{1}{(x^{2}y^{2}z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{8}\int_{-9}^{-4}\int_{-10}^{-7}(\frac{1}{(x^{2}y^{2}z)})dxdydz=\int_{6}^{8}\int_{-9}^{-4}(-\frac{1}{(xy^{2}z)})dydz|_{-10}^{-7}\\&\int_{6}^{8}\int_{-9}^{-4}(\frac{3}{(70y^{2}z)})dydz=\int_{6}^{8}(-\frac{3}{(70yz)})dz|_{-9}^{-4}\\&\int_{6}^{8}(\frac{1}{(168z)})dz=\frac{ln(z)}{168}|_{6}^{8}=1.71\cdot10^{-3}\end{align*}$

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Example Question #332 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{11}\int_{3}^{5.5}\int_{-3}^{-2}(\frac{(25\cdot 3^{(\frac{y}{2})}cos(z + 2))}{(7\cdot 3^{(\frac{x}{2})})})dxdydz\end{align*}$

Possible Answers:

-380.99

-95.25

127

380.99

Correct answer:

380.99

Explanation:

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

$\begin{align*}&\int_{10}^{11}\int_{3}^{5.5}\int_{-3}^{-2}(\frac{(25\cdot 3^{(\frac{y}{2})}cos(z + 2))}{(7\cdot 3^{(\frac{x}{2})})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{11}\int_{3}^{5.5}\int_{-3}^{-2}(\frac{(25\cdot 3^{(\frac{y}{2})}cos(z + 2))}{(7\cdot 3^{(\frac{x}{2})})})dxdydz=\int_{10}^{11}\int_{3}^{5.5}(-\frac{(50\cdot 3^{(\frac{y}{2}-\frac{ x}{2})}cos(z + 2))}{(7ln(3))})dydz|_{-3}^{-2}\\&\int_{10}^{11}\int_{3}^{5.5}(\frac{(150\cdot 3^{(\frac{y}{2})}cos(z + 2)\cdot (3^{(\frac{1}{2})} - 1))}{(7ln(3))})dydz=\int_{10}^{11}(-\frac{(300\cdot 3^{(\frac{y}{2})}\cdot (cos(z + 2) - 3^{(\frac{1}{2})}cos(z + 2)))}{(7ln(3)^{2})})dz|_{3}^{5.5}\\&\int_{10}^{11}(\frac{(900cos(z + 2)\cdot (3^{(\frac{1}{2})} + 9\cdot 3^{(\frac{1}{4})} - 3\cdot 3^{(\frac{3}{4})} - 3))}{(7ln(3)^{2})})dz=\frac{(900sin(z + 2)\cdot (3^{(\frac{1}{2})} + 9\cdot 3^{(\frac{1}{4})} - 3\cdot 3^{(\frac{3}{4})} - 3))}{(7ln(3)^{2})}|_{10}^{11}=380.99\end{align*}$

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Example Question #471 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{10.5}\int_{8}^{12}\int_{-7}^{-3}(\frac{(6xcos(3z)sin(4y))}{29})dxdydz\end{align*}$

Possible Answers:

0.22

0.44

-2.22

-0.22

Correct answer:

0.44

Explanation:

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

$\begin{align*}&\int_{9}^{10.5}\int_{8}^{12}\int_{-7}^{-3}(\frac{(6xcos(3z)sin(4y))}{29})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{10.5}\int_{8}^{12}\int_{-7}^{-3}(\frac{(6xcos(3z)sin(4y))}{29})dxdydz=\int_{9}^{10.5}\int_{8}^{12}(\frac{(3x^{2}cos(3z)sin(4y))}{29})dydz|_{-7}^{-3}\\&\int_{9}^{10.5}\int_{8}^{12}(-\frac{(120cos(3z)sin(4y))}{29})dydz=\int_{9}^{10.5}(\frac{(30cos(4y)cos(3z))}{29})dz|_{8}^{12}\\&\int_{9}^{10.5}(-\frac{(120cos(3z)\cdot (\frac{cos(32)}{4}-\frac{ cos(48)}{4}))}{29})dz=-sin(3z)\cdot (\frac{(10cos(32))}{29}-\frac{ (10cos(48))}{29})|_{9}^{10.5}=0.44\end{align*}$

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Example Question #341 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-2.5}\int_{9}^{12.5}\int_{7}^{10.5}(\frac{(17sin(3y))}{(5xz)})dxdydz\end{align*}$

Possible Answers:

$-8.53\cdot10^{-2}$

-2.05

0.51

0.17

Correct answer:

0.51

Explanation:

$\begin{align*}&\int_{-6}^{-2.5}\int_{9}^{12.5}\int_{7}^{10.5}(\frac{(17sin(3y))}{(5xz)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-2.5}\int_{9}^{12.5}\int_{7}^{10.5}(\frac{(17sin(3y))}{(5xz)})dxdydz=\int_{-6}^{-2.5}\int_{9}^{12.5}(\frac{(17sin(3y)ln(x))}{(5z)})dydz|_{7}^{10.5}\\&\int_{-6}^{-2.5}\int_{9}^{12.5}(\frac{(17sin(3y)ln(\frac{3}{2}))}{(5z)})dydz=\int_{-6}^{-2.5}(-\frac{(17cos(3y)ln(\frac{3}{2}))}{(15z)})dz|_{9}^{12.5}\\&\int_{-6}^{-2.5}(\frac{(17ln(\frac{3}{2})\cdot (\frac{cos(27)}{3}-\frac{ cos(\frac{75}{2})}{3}))}{(5z)})dz=ln(z)\cdot (\frac{(17cos(27)ln(\frac{3}{2}))}{15}-\frac{ (17cos(\frac{75}{2})ln(\frac{3}{2}))}{15})|_{-6}^{-2.5}=0.51\end{align*}$

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Example Question #342 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{12}\int_{-6}^{-4.5}\int_{3.5}^{4.5}(\frac{(2sin(y + 2)e^{(-2x)})}{(3z)})dxdydz\end{align*}$

Possible Answers:

$2.79\cdot10^{-6}$

$1.12\cdot10^{-5}$

$-2.23\cdot10^{-6}$

$-5.58\cdot10^{-5}$

Correct answer:

$1.12\cdot10^{-5}$

Explanation:

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

$\begin{align*}&\int_{9}^{12}\int_{-6}^{-4.5}\int_{3.5}^{4.5}(\frac{(2sin(y + 2)e^{(-2x)})}{(3z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{12}\int_{-6}^{-4.5}\int_{3.5}^{4.5}(\frac{(2sin(y + 2)e^{(-2x)})}{(3z)})dxdydz=\int_{9}^{12}\int_{-6}^{-4.5}(-\frac{(sin(y + 2)e^{(-2x)})}{(3z)})dydz|_{3.5}^{4.5}\\&\int_{9}^{12}\int_{-6}^{-4.5}(\frac{(sin(y + 2)e^{(-9)}\cdot (e^{(2)} - 1))}{(3z)})dydz=\int_{9}^{12}(-\frac{(cos(y + 2)e^{(-9)}\cdot (e^{(2)} - 1))}{(3z)})dz|_{-6}^{-4.5}\\&\int_{9}^{12}(\frac{(e^{(-9)}\cdot (cos(4) - cos(\frac{5}{2}))\cdot (e^{(2)} - 1))}{(3z)})dz=-\frac{(e^{(-9)}ln(z)\cdot (cos(4) - cos(\frac{5}{2}) - cos(4)e^{(2)} + cos(\frac{5}{2})e^{(2)}))}{3}|_{9}^{12}=1.12\cdot10^{-5}\end{align*}$

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Example Question #341 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{1}^{5}\int_{3}^{7}\int_{-4}^{0.5}(\frac{(4\cdot 2^y\cdot 3^{(\frac{x}{4})})}{z})dxdydz\end{align*}$

Possible Answers:

-1651.32

$-1.65\cdot10^{4}$

6605.26

3302.63

Correct answer:

3302.63

Explanation:

$\begin{align*}&\int_{1}^{5}\int_{3}^{7}\int_{-4}^{0.5}(\frac{(4\cdot 2^y\cdot 3^{(\frac{x}{4})})}{z})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{1}^{5}\int_{3}^{7}\int_{-4}^{0.5}(\frac{(4\cdot 2^y\cdot 3^{(\frac{x}{4})})}{z})dxdydz=\int_{1}^{5}\int_{3}^{7}(\frac{(16\cdot 2^y\cdot 3^{(\frac{x}{4})})}{(zln(3))})dydz|_{-4}^{0.5}\\&\int_{1}^{5}\int_{3}^{7}(\frac{(16\cdot 2^y\cdot (3\cdot 3^{(\frac{1}{8})} - 1))}{(3zln(3))})dydz=\int_{1}^{5}(\frac{(16\cdot 2^y\cdot (3\cdot 3^{(\frac{1}{8})} - 1))}{(3zln(2)ln(3))})dz|_{3}^{7}\\&\int_{1}^{5}(\frac{(640\cdot (3\cdot 3^{(\frac{1}{8})} - 1))}{(zln(2)ln(3))})dz=\frac{(ln(z)\cdot (1920\cdot 3^{(\frac{1}{8})} - 640))}{(ln(2)ln(3))}|_{1}^{5}=3302.63\end{align*}$

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Example Question #345 : Triple Integration In Cartesian Coordinates

Integrate:

$\int_{0}^{3}\int_{y}^{2y}\int_{0}^{x}dzdxdy$

Possible Answers:

$\frac{9}{2}$

$\frac{27}{2}$

Correct answer:

$\frac{27}{2}$

Explanation:

To integrate, we work from the inside outward.

The first integral is

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

This becomes the integrand of the next outermost integral, which is

$\int_{y}^{2y}xdx=\frac{x^2}{2}\Big|^{2y}_{y}= \frac{3y^2}{2}$

This becomes the integrand of the outermost integral, which is

$\int_{0}^{3}\frac{3y^2}{2}dy=\frac{y^3}{2}\Big|^{3}_{0}= \frac{27}{2}$

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Example Question #346 : Triple Integration In Cartesian Coordinates

Evaluate:

$\int_{0}^{1}\int_{x}^{2x}\int_{0}^{y}zdzdydx$

Possible Answers:

$\frac{7}{10}$

$\frac{7}{24}$

$\frac{1}{3}$

Correct answer:

$\frac{7}{24}$

Explanation:

To integrate, we work from the inside outwards.

The first integral we perform is

$\int_{0}^{y}zdz= \frac{z^2}{2}\Big|^{y}_{0}= \frac{y^2}{2}$

This becomes the integrand for the next integral:

$\int_{x}^{2x}\frac{y^2}{2}dy=\frac{y^3}{6}\Big|^{2x}_{x}= \frac{8x^3-x^3}{6}=\frac{7x^3}{6}$

This becomes the integrand for the last integral:

$\int_{0}^{1}\frac{7x^3}{6}dx=\frac{7x^4}{24}\Big|^{1}_{0}= \frac{7}{24}$

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Example Question #347 : Triple Integration In Cartesian Coordinates

Evaluate

$\int_0^1\int_0^1\int_0^{2x^2+1}(xyz)dydxdz$

Possible Answers:

$\frac{7}{23}$

$\frac{11}{12}$

$\frac{11}{43}$

$\frac{13}{48}$

$\frac{11}{7}$

Correct answer:

$\frac{13}{48}$

Explanation:

$\int_0^1\int_0^1\int_0^{2x^2+1}(xyz)dydxdz$

First notice the order of integration. We start by integrating with respect to , and we treat and as constants.

$\int_0^1\int_0^1\left[xz\frac{1}{2}y^2\right]_{y=0}^{y=2x^2+1}dxdz$

$=\int_0^1\int_0^1(xz)\left[\frac{1}{2}(2x^2+1)^2\right]dxdz$

Expand the integrand and then integrate with respect to term-by-term:

$xz\left[\frac{1}{2}(2x^2+1)^2\right]=xz\left[\frac{1}{2}(4x^4+4x^2+1) \right ]=z\frac{1}{2}\left[4x^5+4x^3+x \right ]$

$=\frac{1}{2} \int_0^1\int_0^1z\left[4x^5+4x^3+x \right ] dxdz$

$=\frac{1}{2}\int_0^1z\left[\frac{2}{3}x^6 +x^4+\frac{1}{2}x^2 \right ]_{x=0}^{x=1} dz$

$=\frac{1}{2}\int_0^1\frac{13}{12}zdz$

$=\frac{13}{24}\int_0^1zdz$

$\frac{13}{24}\left[\frac{1}{2}z^2 \right ]_{z=0}^{z=1}=\frac{13}{48}$

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Calculus 3 : Multiple Integration

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #471 : Triple Integrals

Example Question #471 : Multiple Integration

Example Question #332 : Triple Integration In Cartesian Coordinates

Example Question #471 : Triple Integrals

Example Question #341 : Triple Integration In Cartesian Coordinates

Example Question #342 : Triple Integration In Cartesian Coordinates

Example Question #341 : Triple Integration In Cartesian Coordinates

Example Question #345 : Triple Integration In Cartesian Coordinates

Example Question #346 : Triple Integration In Cartesian Coordinates

Example Question #347 : Triple Integration In Cartesian Coordinates

All Calculus 3 Resources

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