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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

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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Example Question #348 : Triple Integration In Cartesian Coordinates

Evaluate the following integral

$\int \int \int_D 15xyz \ dV$ , where $D=[1,2]\times[3,4]\times [5,6]$

Possible Answers:

$\frac{3465}{8}$

$\frac{1}{8}$

$\frac{65}{8}$

Correct answer:

$\frac{3465}{8}$

Explanation:

$\int_{1}^{2} \int_{3}^{4} \int_{5}^{6} 15xyz \ dx\ dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} \frac{15}{2}x^2yz\Big|_{5}^{6} dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} (\frac{15}{2}(6)^2yz)-(\frac{15}{2}(5)^2yz) dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} \frac{165}{2}yz\ dy\ dz$

$=\int_{1}^{2} \frac{165}{4}y^2z\Big|_{3}^{4}\ dz$

$=\int_{1}^{2} \frac{165}{4}(4)^2z-\frac{165}{4}(3)^2z\ dz$

$=\int_{1}^{2} \frac{1155}{4}z\ dz$

$=\frac{1155}{8}z^2\Big|_{1}^{2}$

$=\frac{1155}{8}(2)^2-\frac{1155}{8}(1)^2=\frac{3465}{8}$

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Example Question #341 : Triple Integration In Cartesian Coordinates

Evaluate the following integral

$\int \int \int_D 15xyz \ dV$ , where $D=[1,2]\times[3,4]\times [5,6]$

Possible Answers:

$\frac{3465}{8}$

$\frac{1}{8}$

$\frac{65}{8}$

Correct answer:

$\frac{3465}{8}$

Explanation:

$\int_{1}^{2} \int_{3}^{4} \int_{5}^{6} 15xyz \ dx\ dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} \frac{15}{2}x^2yz\Big|_{5}^{6} dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} (\frac{15}{2}(6)^2yz)-(\frac{15}{2}(5)^2yz) dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} \frac{165}{2}yz\ dy\ dz$

$=\int_{1}^{2} \frac{165}{4}y^2z\Big|_{3}^{4}\ dz$

$=\int_{1}^{2} \frac{165}{4}(4)^2z-\frac{165}{4}(3)^2z\ dz$

$=\int_{1}^{2} \frac{1155}{4}z\ dz$

$=\frac{1155}{8}z^2\Big|_{1}^{2}$

$=\frac{1155}{8}(2)^2-\frac{1155}{8}(1)^2=\frac{3465}{8}$

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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 #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

Example Question #348 : Triple Integration In Cartesian Coordinates

Example Question #341 : Triple Integration In Cartesian Coordinates

All Calculus 3 Resources

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