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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #431 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{10}\int_{5}^{9.5}\int_{-10}^{-8}(\frac{(6z^{2}sin(x + 1)e^{(-y)})}{49})dxdydz\end{align*}$

Possible Answers:

0.18

-1.77

2.13

-0.35

Correct answer:

-0.35

Explanation:

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

$\begin{align*}&\int_{6}^{10}\int_{5}^{9.5}\int_{-10}^{-8}(\frac{(6z^{2}sin(x + 1)e^{(-y)})}{49})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{10}\int_{5}^{9.5}\int_{-10}^{-8}(\frac{(6z^{2}sin(x + 1)e^{(-y)})}{49})dxdydz=\int_{6}^{10}\int_{5}^{9.5}(-\frac{(6z^{2}cos(x + 1)e^{(-y)})}{49})dydz|_{-10}^{-8}\\&\int_{6}^{10}\int_{5}^{9.5}(-\frac{(6z^{2}e^{(-y)}\cdot (cos(7) - cos(9)))}{49})dydz=\int_{6}^{10}(\frac{(6z^{2}e^{(-y)}\cdot (cos(7) - cos(9)))}{49})dz|_{5}^{9.5}\\&\int_{6}^{10}(-\frac{(6z^{2}e^{(-\frac{19}{2})}\cdot (cos(7) - cos(9))\cdot (e^{(\frac{9}{2})} - 1))}{49})dz=-z^{3}\cdot (\frac{(2cos(7)e^{(-5)})}{49}-\frac{ (2cos(9)e^{(-5)})}{49}-\frac{ (2cos(7)e^{(-\frac{19}{2})})}{49}+\frac{ (2cos(9)e^{(-\frac{19}{2})})}{49})|_{6}^{10}=-0.35\end{align*}$

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Example Question #432 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{7}\int_{7}^{9.5}\int_{-6}^{-2.5}(\frac{(3y^{2}cos(x + 1)e^{(-2z)})}{53})dxdydz\end{align*}$

Possible Answers:

-0.12

-0.09

0.09

-0.02

Correct answer:

-0.02

Explanation:

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

$\begin{align*}&\int_{3}^{7}\int_{7}^{9.5}\int_{-6}^{-2.5}(\frac{(3y^{2}cos(x + 1)e^{(-2z)})}{53})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{7}\int_{7}^{9.5}\int_{-6}^{-2.5}(\frac{(3y^{2}cos(x + 1)e^{(-2z)})}{53})dxdydz=\int_{3}^{7}\int_{7}^{9.5}(\frac{(3y^{2}sin(x + 1)e^{(-2z)})}{53})dydz|_{-6}^{-2.5}\\&\int_{3}^{7}\int_{7}^{9.5}(-\frac{(3y^{2}e^{(-2z)}\cdot (sin(\frac{3}{2}) - sin(5)))}{53})dydz=\int_{3}^{7}(-y^{3}\cdot (\frac{(e^{(-2z)}sin(\frac{3}{2}))}{53}-\frac{ (e^{(-2z)}sin(5))}{53}))dz|_{7}^{9.5}\\&\int_{3}^{7}(-\frac{(4115e^{(-2z)}\cdot (sin(\frac{3}{2}) - sin(5)))}{424})dz=\frac{(4115e^{(-2z)}\cdot (sin(\frac{3}{2}) - sin(5)))}{848}|_{3}^{7}=-0.02\end{align*}$

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Example Question #431 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-6.5}\int_{6}^{11}\int_{-3}^{-0.5}(\frac{(57e^{(2x)})}{(5y^{2})})dxdydz\end{align*}$

Possible Answers:

-0.1

0.1

-1.18

0.39

Correct answer:

0.39

Explanation:

$\begin{align*}&\int_{-9}^{-6.5}\int_{6}^{11}\int_{-3}^{-0.5}(\frac{(57e^{(2x)})}{(5y^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-6.5}\int_{6}^{11}\int_{-3}^{-0.5}(\frac{(57e^{(2x)})}{(5y^{2})})dxdydz=\int_{-9}^{-6.5}\int_{6}^{11}(\frac{(57e^{(2x)})}{(10y^{2})})dydz|_{-3}^{-0.5}\\&\int_{-9}^{-6.5}\int_{6}^{11}(\frac{(57e^{(-6)}\cdot (e^{(5)} - 1))}{(10y^{2})})dydz=\int_{-9}^{-6.5}(-\frac{(57e^{(-6)}\cdot (e^{(5)} - 1))}{(10y)})dz|_{6}^{11}\\&\int_{-9}^{-6.5}(\frac{(19e^{(-6)}\cdot (e^{(5)} - 1))}{44})dz=\frac{(19ze^{(-6)}\cdot (e^{(5)} - 1))}{44}|_{-9}^{-6.5}=0.39\end{align*}$

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Example Question #434 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{11.5}\int_{-6}^{-4}\int_{-8}^{-3.5}(\frac{(5sin(4y)sin(z + 2))}{x})dxdydz\end{align*}$

Possible Answers:

0.07

-1.42

1.07

-0.36

Correct answer:

-0.36

Explanation:

$\begin{align*}&\int_{10}^{11.5}\int_{-6}^{-4}\int_{-8}^{-3.5}(\frac{(5sin(4y)sin(z + 2))}{x})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{11.5}\int_{-6}^{-4}\int_{-8}^{-3.5}(\frac{(5sin(4y)sin(z + 2))}{x})dxdydz=\int_{10}^{11.5}\int_{-6}^{-4}(5sin(4y)sin(z + 2)ln(x))dydz|_{-8}^{-3.5}\\&\int_{10}^{11.5}\int_{-6}^{-4}(5sin(4y)sin(z + 2)ln(\frac{7}{16}))dydz=\int_{10}^{11.5}(-\frac{(5cos(4y)sin(z + 2)ln(\frac{7}{16}))}{4})dz|_{-6}^{-4}\\&\int_{10}^{11.5}(-5sin(z + 2)ln(\frac{7}{16})\cdot (\frac{cos(16)}{4}-\frac{ cos(24)}{4}))dz=5cos(z + 2)ln(\frac{7}{16})\cdot (\frac{cos(16)}{4}-\frac{ cos(24)}{4})|_{10}^{11.5}=-0.36\end{align*}$

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Example Question #431 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{8}^{10}\int_{-9}^{-8}\int_{7}^{10.5}(\frac{(23y^{2}cos(4z))}{(5x)})dxdydz\end{align*}$

Possible Answers:

-1.09

-26.13

6.53

19.6

Correct answer:

6.53

Explanation:

$\begin{align*}&\int_{8}^{10}\int_{-9}^{-8}\int_{7}^{10.5}(\frac{(23y^{2}cos(4z))}{(5x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{8}^{10}\int_{-9}^{-8}\int_{7}^{10.5}(\frac{(23y^{2}cos(4z))}{(5x)})dxdydz=\int_{8}^{10}\int_{-9}^{-8}(\frac{(23y^{2}cos(4z)ln(x))}{5})dydz|_{7}^{10.5}\\&\int_{8}^{10}\int_{-9}^{-8}(\frac{(23y^{2}cos(4z)ln(\frac{3}{2}))}{5})dydz=\int_{8}^{10}(\frac{(23y^{3}cos(4z)ln(\frac{3}{2}))}{15})dz|_{-9}^{-8}\\&\int_{8}^{10}(\frac{(4991cos(4z)ln(\frac{3}{2}))}{15})dz=\frac{(4991sin(4z)ln(\frac{3}{2}))}{60}|_{8}^{10}=6.53\end{align*}$

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Example Question #435 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-10}^{-7.5}\int_{7}^{9}\int_{-3.5}^{1}(\frac{(sin(y + 1)e^{(2x)})}{(92z)})dxdydz\end{align*}$

Possible Answers:

-0.05

0.91

-0.09

-0.01

Correct answer:

-0.01

Explanation:

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

$\begin{align*}&\int_{-10}^{-7.5}\int_{7}^{9}\int_{-3.5}^{1}(\frac{(sin(y + 1)e^{(2x)})}{(92z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-10}^{-7.5}\int_{7}^{9}\int_{-3.5}^{1}(\frac{(sin(y + 1)e^{(2x)})}{(92z)})dxdydz=\int_{-10}^{-7.5}\int_{7}^{9}(\frac{(sin(y + 1)e^{(2x)})}{(184z)})dydz|_{-3.5}^{1}\\&\int_{-10}^{-7.5}\int_{7}^{9}(\frac{(sin(y + 1)e^{(-7)}\cdot (e^{(9)} - 1))}{(184z)})dydz=\int_{-10}^{-7.5}(-\frac{(cos(y + 1)e^{(-7)}\cdot (e^{(9)} - 1))}{(184z)})dz|_{7}^{9}\\&\int_{-10}^{-7.5}(\frac{(e^{(-7)}\cdot (cos(8) - cos(10))\cdot (e^{(9)} - 1))}{(184z)})dz=-\frac{(e^{(-7)}ln(z)\cdot (cos(8) - cos(10) - cos(8)e^{(9)} + cos(10)e^{(9)}))}{184}|_{-10}^{-7.5}=-0.01\end{align*}$

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Example Question #436 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3.5}^{-0.5}\int_{-10}^{-5}\int_{-3.5}^{1.5}(19e^{(-x)}e^{(z)})dxdydz\end{align*}$

Possible Answers:

1801

900.5

-450.2

Correct answer:

1801

Explanation:

$\begin{align*}&\int_{-3.5}^{-0.5}\int_{-10}^{-5}\int_{-3.5}^{1.5}(19e^{(-x)}e^{(z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3.5}^{-0.5}\int_{-10}^{-5}\int_{-3.5}^{1.5}(19e^{(-x)}e^{(z)})dxdydz=\int_{-3.5}^{-0.5}\int_{-10}^{-5}(-19e^{(z - x)})dydz|_{-3.5}^{1.5}\\&\int_{-3.5}^{-0.5}\int_{-10}^{-5}(19e^{(-\frac{3}{2})}e^{(z)}\cdot (e^{(5)} - 1))dydz=\int_{-3.5}^{-0.5}(19ye^{(-\frac{3}{2})}e^{(z)}\cdot (e^{(5)} - 1))dz|_{-10}^{-5}\\&\int_{-3.5}^{-0.5}(95e^{(-\frac{3}{2})}e^{(z)}\cdot (e^{(5)} - 1))dz=95e^{(-\frac{3}{2})}e^{(z)}\cdot (e^{(5)} - 1)|_{-3.5}^{-0.5}=1801\end{align*}$

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Example Question #441 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-5}\int_{3}^{7}\int_{-4.5}^{-1.5}(\frac{(5z^{2}e^{(2y)})}{(47\cdot 2^x)})dxdydz\end{align*}$

Possible Answers:

$-1.103\cdot10^{9}$

$3.677\cdot10^{8}$

$2.206\cdot10^{9}$

$-1.839\cdot10^{8}$

Correct answer:

$3.677\cdot10^{8}$

Explanation:

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

$\begin{align*}&\int_{-9}^{-5}\int_{3}^{7}\int_{-4.5}^{-1.5}(\frac{(5z^{2}e^{(2y)})}{(47\cdot 2^x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-5}\int_{3}^{7}\int_{-4.5}^{-1.5}(\frac{(5z^{2}e^{(2y)})}{(47\cdot 2^x)})dxdydz=\int_{-9}^{-5}\int_{3}^{7}(-\frac{(5z^{2}e^{(2y)})}{(47\cdot 2^xln(2))})dydz|_{-4.5}^{-1.5}\\&\int_{-9}^{-5}\int_{3}^{7}(\frac{(70\cdot 2^{(\frac{1}{2})}z^{2}e^{(2y)})}{(47ln(2))})dydz=\int_{-9}^{-5}(\frac{(35\cdot 2^{(\frac{1}{2})}z^{2}e^{(2y)})}{(47ln(2))})dz|_{3}^{7}\\&\int_{-9}^{-5}(\frac{(35\cdot 2^{(\frac{1}{2})}z^{2}e^{(6)}\cdot (e^{(8)} - 1))}{(47ln(2))})dz=-\frac{(z^{3}\cdot (35\cdot 2^{(\frac{1}{2})}e^{(6)} - 35\cdot 2^{(\frac{1}{2})}e^{(14)}))}{(141ln(2))}|_{-9}^{-5}=3.677\cdot10^{8}\end{align*}$

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Example Question #441 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{2}\int_{8}^{11.5}\int_{10}^{11}(\frac{(3ye^{(-z)})}{(43x)})dxdydz\end{align*}$

Possible Answers:

-13.58

0.755

4.53

-1.51

Correct answer:

4.53

Explanation:

$\begin{align*}&\int_{-3}^{2}\int_{8}^{11.5}\int_{10}^{11}(\frac{(3ye^{(-z)})}{(43x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{2}\int_{8}^{11.5}\int_{10}^{11}(\frac{(3ye^{(-z)})}{(43x)})dxdydz=\int_{-3}^{2}\int_{8}^{11.5}(\frac{(3ye^{(-z)}ln(x))}{43})dydz|_{10}^{11}\\&\int_{-3}^{2}\int_{8}^{11.5}(\frac{(3ye^{(-z)}ln(\frac{11}{10}))}{43})dydz=\int_{-3}^{2}(\frac{(3y^{2}e^{(-z)}ln(\frac{11}{10}))}{86})dz|_{8}^{11.5}\\&\int_{-3}^{2}(\frac{(819e^{(-z)}ln(\frac{11}{10}))}{344})dz=-\frac{(819e^{(-z)}ln(\frac{11}{10}))}{344}|_{-3}^{2}=4.53\end{align*}$

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Example Question #911 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{7.5}\int_{10}^{15}\int_{8}^{11.5}(\frac{(5sin(x + 1))}{(33yz)})dxdydz\end{align*}$

Possible Answers:

-0.215

0.0215

-0.107

0.322

Correct answer:

-0.107

Explanation:

$\begin{align*}&\int_{3}^{7.5}\int_{10}^{15}\int_{8}^{11.5}(\frac{(5sin(x + 1))}{(33yz)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{7.5}\int_{10}^{15}\int_{8}^{11.5}(\frac{(5sin(x + 1))}{(33yz)})dxdydz=\int_{3}^{7.5}\int_{10}^{15}(-\frac{(5cos(x + 1))}{(33yz)})dydz|_{8}^{11.5}\\&\int_{3}^{7.5}\int_{10}^{15}(\frac{(5\cdot (cos(9) - cos(\frac{25}{2})))}{(33yz)})dydz=\int_{3}^{7.5}(\frac{(ln(y)\cdot (5cos(9) - 5cos(\frac{25}{2})))}{(33z)})dz|_{10}^{15}\\&\int_{3}^{7.5}(\frac{(5ln(\frac{3}{2})\cdot (cos(9) - cos(\frac{25}{2})))}{(33z)})dz=ln(z)\cdot (\frac{(5cos(9)ln(\frac{3}{2}))}{33}-\frac{ (5cos(\frac{25}{2})ln(\frac{3}{2}))}{33})|_{3}^{7.5}=-0.107\end{align*}$

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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 #431 : Multiple Integration

Example Question #432 : Multiple Integration

Example Question #431 : Triple Integrals

Example Question #434 : Multiple Integration

Example Question #431 : Multiple Integration

Example Question #435 : Multiple Integration

Example Question #436 : Multiple Integration

Example Question #441 : Multiple Integration

Example Question #441 : Multiple Integration

Example Question #911 : Calculus 3

All Calculus 3 Resources

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