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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #291 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{10}\int_{4}^{8.5}\int_{-9}^{-4.5}(\frac{(2cos(z + 1)e^{(y)})}{61})dxdydz\end{align*}$

Possible Answers:

8559

-356.6

237.8

Correct answer:

Explanation:

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

$\begin{align*}&\int_{7}^{10}\int_{4}^{8.5}\int_{-9}^{-4.5}(\frac{(2cos(z + 1)e^{(y)})}{61})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{10}\int_{4}^{8.5}\int_{-9}^{-4.5}(\frac{(2cos(z + 1)e^{(y)})}{61})dxdydz=\int_{7}^{10}\int_{4}^{8.5}(\frac{(2xcos(z + 1)e^{(y)})}{61})dydz|_{-9}^{-4.5}\\&\int_{7}^{10}\int_{4}^{8.5}(\frac{(9cos(z + 1)e^{(y)})}{61})dydz=\int_{7}^{10}(\frac{(9cos(z + 1)e^{(y)})}{61})dz|_{4}^{8.5}\\&\int_{7}^{10}(-\frac{(9cos(z + 1)\cdot (e^{(4)} - e^{(\frac{17}{2})}))}{61})dz=\frac{(9sin(z + 1)e^{(4)}\cdot (e^{(\frac{9}{2})} - 1))}{61}|_{7}^{10}=-1427\end{align*}$

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Example Question #421 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-7}^{-3}\int_{-6}^{-4}\int_{4.5}^{5.5}(\frac{(10cos(y + 1)cos(z + 1)e^{(2x)})}{31})dxdydz\end{align*}$

Possible Answers:

2184

10922

Correct answer:

10922

Explanation:

$\begin{align*}&\int_{-7}^{-3}\int_{-6}^{-4}\int_{4.5}^{5.5}(\frac{(10cos(y + 1)cos(z + 1)e^{(2x)})}{31})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-7}^{-3}\int_{-6}^{-4}\int_{4.5}^{5.5}(\frac{(10cos(y + 1)cos(z + 1)e^{(2x)})}{31})dxdydz=\int_{-7}^{-3}\int_{-6}^{-4}(\frac{(5cos(y + 1)cos(z + 1)e^{(2x)})}{31})dydz|_{4.5}^{5.5}\\&\int_{-7}^{-3}\int_{-6}^{-4}(\frac{(5cos(y + 1)cos(z + 1)e^{(9)}\cdot (e^{(2)} - 1))}{31})dydz=\int_{-7}^{-3}(\frac{(5cos(z + 1)sin(y + 1)e^{(9)}\cdot (e^{(2)} - 1))}{31})dz|_{-6}^{-4}\\&\int_{-7}^{-3}(-\frac{(5cos(z + 1)e^{(9)}\cdot (sin(3) - sin(5))\cdot (e^{(2)} - 1))}{31})dz=-\frac{(5sin(z + 1)e^{(9)}\cdot (sin(3) - sin(5))\cdot (e^{(2)} - 1))}{31}|_{-7}^{-3}=10922\end{align*}$

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Example Question #422 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3.5}^{1}\int_{3}^{6.5}\int_{7}^{11}(14\cdot 2^{(\frac{y}{2})}\cdot 2^zsin(4x))dxdydz\end{align*}$

Possible Answers:

91.35

1462

-365.4

Correct answer:

-365.4

Explanation:

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

$\begin{align*}&\int_{-3.5}^{1}\int_{3}^{6.5}\int_{7}^{11}(14\cdot 2^{(\frac{y}{2})}\cdot 2^zsin(4x))dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3.5}^{1}\int_{3}^{6.5}\int_{7}^{11}(14\cdot 2^{(\frac{y}{2})}\cdot 2^zsin(4x))dxdydz=\int_{-3.5}^{1}\int_{3}^{6.5}(-\frac{(7\cdot 2^{(\frac{y}{2}+ z)}cos(4x))}{2})dydz|_{7}^{11}\\&\int_{-3.5}^{1}\int_{3}^{6.5}(2^{(\frac{y}{2}+ z)}\cdot (\frac{(7cos(28))}{2}-\frac{ (7cos(44))}{2}))dydz=\int_{-3.5}^{1}(\frac{(7\cdot 2^{(\frac{y}{2}+ z)}\cdot (cos(28) - cos(44)))}{ln(2)})dz|_{3}^{6.5}\\&\int_{-3.5}^{1}(-\frac{(14\cdot 2^{(\frac{1}{4})}\cdot 2^z\cdot (cos(28) - cos(44))\cdot (2^{(\frac{1}{4})} - 4))}{ln(2)})dz=-\frac{(14\cdot 2^z\cdot (2^{(\frac{1}{2})}cos(28) - 4\cdot 2^{(\frac{1}{4})}cos(28) - 2^{(\frac{1}{2})}cos(44) + 4\cdot 2^{(\frac{1}{4})}cos(44)))}{ln(2)^{2}}|_{-3.5}^{1}=-365.4\end{align*}$

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Example Question #422 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{-2}\int_{8}^{10}\int_{3}^{7.5}(\frac{(3^{(\frac{x}{3})}\cdot 3^{(\frac{z}{2})})}{y})dxdydz\end{align*}$

Possible Answers:

1.97

0.33

-0.33

-3.94

Correct answer:

1.97

Explanation:

$\begin{align*}&\int_{-3}^{-2}\int_{8}^{10}\int_{3}^{7.5}(\frac{(3^{(\frac{x}{3})}\cdot 3^{(\frac{z}{2})})}{y})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{-2}\int_{8}^{10}\int_{3}^{7.5}(\frac{(3^{(\frac{x}{3})}\cdot 3^{(\frac{z}{2})})}{y})dxdydz=\int_{-3}^{-2}\int_{8}^{10}(\frac{(3\cdot 3^{(\frac{x}{3}+\frac{ z}{2})})}{(yln(3))})dydz|_{3}^{7.5}\\&\int_{-3}^{-2}\int_{8}^{10}(\frac{(9\cdot 3^{(\frac{z}{2})}\cdot (3\cdot 3^{(\frac{1}{2})} - 1))}{(yln(3))})dydz=\int_{-3}^{-2}(-\frac{(ln(y)\cdot (9\cdot 3^{(\frac{z}{2})} - 27\cdot 3^{(\frac{1}{2})}\cdot 3^{(\frac{z}{2})}))}{ln(3)})dz|_{8}^{10}\\&\int_{-3}^{-2}(\frac{(9\cdot 3^{(\frac{z}{2})}ln(\frac{5}{4})\cdot (3\cdot 3^{(\frac{1}{2})} - 1))}{ln(3)})dz=-\frac{(18\cdot 3^{(\frac{z}{2})}\cdot (ln(\frac{5}{4}) - 3\cdot 3^{(\frac{1}{2})}ln(\frac{5}{4})))}{ln(3)^{2}}|_{-3}^{-2}=1.97\end{align*}$

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Example Question #423 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-8}^{-4.5}\int_{8}^{9}\int_{-4}^{-2}(5y^{2}cos(z + 1)e^{(2x)})dxdydz\end{align*}$

Possible Answers:

-13.11

3.28

-0.82

16.38

Correct answer:

3.28

Explanation:

$\begin{align*}&\int_{-8}^{-4.5}\int_{8}^{9}\int_{-4}^{-2}(5y^{2}cos(z + 1)e^{(2x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-8}^{-4.5}\int_{8}^{9}\int_{-4}^{-2}(5y^{2}cos(z + 1)e^{(2x)})dxdydz=\int_{-8}^{-4.5}\int_{8}^{9}(\frac{(5y^{2}cos(z + 1)e^{(2x)})}{2})dydz|_{-4}^{-2}\\&\int_{-8}^{-4.5}\int_{8}^{9}(\frac{(5y^{2}cos(z + 1)e^{(-8)}\cdot (e^{(4)} - 1))}{2})dydz=\int_{-8}^{-4.5}(\frac{(5y^{3}cos(z + 1)e^{(-8)}\cdot (e^{(4)} - 1))}{6})dz|_{8}^{9}\\&\int_{-8}^{-4.5}(\frac{(1085cos(z + 1)e^{(-8)}\cdot (e^{(4)} - 1))}{6})dz=\frac{(1085sin(z + 1)e^{(-8)}\cdot (e^{(4)} - 1))}{6}|_{-8}^{-4.5}=3.28\end{align*}$

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Example Question #291 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-10}^{-5}\int_{7}^{11}\int_{-9}^{-6.5}(\frac{(8cos(x + 2)sin(y + 1))}{(207z)})dxdydz\end{align*}$

Possible Answers:

-0.01

-0.04

0.04

0.17

Correct answer:

0.04

Explanation:

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

$\begin{align*}&\int_{-10}^{-5}\int_{7}^{11}\int_{-9}^{-6.5}(\frac{(8cos(x + 2)sin(y + 1))}{(207z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-10}^{-5}\int_{7}^{11}\int_{-9}^{-6.5}(\frac{(8cos(x + 2)sin(y + 1))}{(207z)})dxdydz=\int_{-10}^{-5}\int_{7}^{11}(\frac{(8sin(x + 2)sin(y + 1))}{(207z)})dydz|_{-9}^{-6.5}\\&\int_{-10}^{-5}\int_{7}^{11}(\frac{(8sin(y + 1)\cdot (sin(7) - sin(\frac{9}{2})))}{(207z)})dydz=\int_{-10}^{-5}(-\frac{(8cos(y + 1)\cdot (sin(7) - sin(\frac{9}{2})))}{(207z)})dz|_{7}^{11}\\&\int_{-10}^{-5}(\frac{(8\cdot (cos(8) - cos(12))\cdot (sin(7) - sin(\frac{9}{2})))}{(207z)})dz=ln(z)\cdot (\frac{(8cos(8)sin(7))}{207}-\frac{ (8cos(8)sin(\frac{9}{2}))}{207}-\frac{ (8cos(12)sin(7))}{207}+\frac{ (8cos(12)sin(\frac{9}{2}))}{207})|_{-10}^{-5}=0.04\end{align*}$

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Example Question #292 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{10}^{14}\int_{-6}^{-1}\int_{7}^{11}(\frac{(11cos(y + 1)sin(z + 1))}{(2x)})dxdydz\end{align*}$

Possible Answers:

7.29

-3.64

0.46

-1.82

Correct answer:

-1.82

Explanation:

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

$\begin{align*}&\int_{10}^{14}\int_{-6}^{-1}\int_{7}^{11}(\frac{(11cos(y + 1)sin(z + 1))}{(2x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{10}^{14}\int_{-6}^{-1}\int_{7}^{11}(\frac{(11cos(y + 1)sin(z + 1))}{(2x)})dxdydz=\int_{10}^{14}\int_{-6}^{-1}(\frac{(11cos(y + 1)sin(z + 1)ln(x))}{2})dydz|_{7}^{11}\\&\int_{10}^{14}\int_{-6}^{-1}(\frac{(11cos(y + 1)sin(z + 1)ln(\frac{11}{7}))}{2})dydz=\int_{10}^{14}(\frac{(11sin(y + 1)sin(z + 1)ln(\frac{11}{7}))}{2})dz|_{-6}^{-1}\\&\int_{10}^{14}(\frac{(11sin(z + 1)ln(\frac{11}{7})sin(5))}{2})dz=-\frac{(11cos(z + 1)ln(\frac{11}{7})sin(5))}{2}|_{10}^{14}=-1.82\end{align*}$

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Example Question #431 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{10.5}\int_{-7}^{-3.5}\int_{-4}^{-1}(\frac{(23e^{(-2x)})}{(9yz^{2})})dxdydz\end{align*}$

Possible Answers:

41.8

20.9

-41.8

-250.8

Correct answer:

-41.8

Explanation:

$\begin{align*}&\int_{9}^{10.5}\int_{-7}^{-3.5}\int_{-4}^{-1}(\frac{(23e^{(-2x)})}{(9yz^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{10.5}\int_{-7}^{-3.5}\int_{-4}^{-1}(\frac{(23e^{(-2x)})}{(9yz^{2})})dxdydz=\int_{9}^{10.5}\int_{-7}^{-3.5}(-\frac{(23e^{(-2x)})}{(18yz^{2})})dydz|_{-4}^{-1}\\&\int_{9}^{10.5}\int_{-7}^{-3.5}(\frac{(23e^{(2)}\cdot (e^{(6)} - 1))}{(18yz^{2})})dydz=\int_{9}^{10.5}(\frac{(23e^{(2)}ln(y)\cdot (e^{(6)} - 1))}{(18z^{2})})dz|_{-7}^{-3.5}\\&\int_{9}^{10.5}(-\frac{(23e^{(2)}ln(2)\cdot (e^{(6)} - 1))}{(18z^{2})})dz=\frac{(23e^{(2)}ln(2)\cdot (e^{(6)} - 1))}{(18z)}|_{9}^{10.5}=-41.8\end{align*}$

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Example Question #432 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{8}^{11.5}\int_{-3.5}^{-1.5}\int_{3}^{5}(\frac{(4\cdot 2^{(\frac{x}{4})}\cdot 3^ycos(3z))}{3})dxdydz\end{align*}$

Possible Answers:

0.54

-1.34

-0.07

0.27

Correct answer:

0.27

Explanation:

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

$\begin{align*}&\int_{8}^{11.5}\int_{-3.5}^{-1.5}\int_{3}^{5}(\frac{(4\cdot 2^{(\frac{x}{4})}\cdot 3^ycos(3z))}{3})dxdydz\\&\text{The approach is to take it step by step:}\\&\int_{8}^{11.5}\int_{-3.5}^{-1.5}\int_{3}^{5}(\frac{(4\cdot 2^{(\frac{x}{4})}\cdot 3^ycos(3z))}{3})dxdydz=\int_{8}^{11.5}\int_{-3.5}^{-1.5}(\frac{(16\cdot 2^{(\frac{x}{4})}\cdot 3^{(y - 1)}cos(3z))}{ln(2)})dydz|_{3}^{5}\\&\int_{8}^{11.5}\int_{-3.5}^{-1.5}(\frac{(4\cdot 3^ycos(3z)\cdot (8\cdot 2^{(\frac{1}{4})} - 4\cdot 2^{(\frac{3}{4})}))}{(3ln(2))})dydz=\int_{8}^{11.5}(\frac{(16\cdot 3^y\cdot (2\cdot 2^{(\frac{1}{4})}cos(3z) - 8^{(\frac{1}{4})}cos(3z)))}{(3ln(2)ln(3))})dz|_{-3.5}^{-1.5}\\&\int_{8}^{11.5}(\frac{(256\cdot 2^{(\frac{1}{4})}\cdot 3^{(\frac{1}{2})}cos(3z) - 128\cdot 2^{(\frac{3}{4})}\cdot 3^{(\frac{1}{2})}cos(3z))}{(243ln(2)ln(3))})dz=\frac{(256\cdot 2^{(\frac{1}{4})}\cdot 3^{(\frac{1}{2})}sin(3z) - 128\cdot 3^{(\frac{1}{2})}\cdot 8^{(\frac{1}{4})}sin(3z))}{(729ln(2)ln(3))}|_{8}^{11.5}=0.27\end{align*}$

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Example Question #433 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{4.5}^{8}\int_{-7}^{-5.5}\int_{-6}^{-4.5}(\frac{(18\cdot 3^zsin(x + 2)sin(3y))}{11})dxdydz\end{align*}$

Possible Answers:

-72.73

12.12

72.73

-12.12

Correct answer:

72.73

Explanation:

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

$\begin{align*}&\int_{4.5}^{8}\int_{-7}^{-5.5}\int_{-6}^{-4.5}(\frac{(18\cdot 3^zsin(x + 2)sin(3y))}{11})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4.5}^{8}\int_{-7}^{-5.5}\int_{-6}^{-4.5}(\frac{(18\cdot 3^zsin(x + 2)sin(3y))}{11})dxdydz=\int_{4.5}^{8}\int_{-7}^{-5.5}(-\frac{(18\cdot 3^zcos(x + 2)sin(3y))}{11})dydz|_{-6}^{-4.5}\\&\int_{4.5}^{8}\int_{-7}^{-5.5}(\frac{(18\cdot 3^zsin(3y)\cdot (cos(4) - cos(\frac{5}{2})))}{11})dydz=\int_{4.5}^{8}(-\frac{(6\cdot 3^zcos(3y)\cdot (cos(4) - cos(\frac{5}{2})))}{11})dz|_{-7}^{-5.5}\\&\int_{4.5}^{8}(\frac{(6\cdot 3^z\cdot (cos(4) - cos(\frac{5}{2}))\cdot (cos(21) - cos(\frac{33}{2})))}{11})dz=\frac{(6\cdot 3^z\cdot (cos(4)cos(21) - cos(\frac{5}{2})cos(21) - cos(4)cos(\frac{33}{2}) + cos(\frac{5}{2})cos(\frac{33}{2})))}{(11ln(3))}|_{4.5}^{8}=72.73\end{align*}$

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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 #291 : Triple Integration In Cartesian Coordinates

Example Question #421 : Multiple Integration

Example Question #422 : Multiple Integration

Example Question #422 : Multiple Integration

Example Question #423 : Multiple Integration

Example Question #291 : Triple Integration In Cartesian Coordinates

Example Question #292 : Triple Integration In Cartesian Coordinates

Example Question #431 : Triple Integrals

Example Question #432 : Triple Integrals

Example Question #433 : Triple Integrals

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