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Calculus 3 : Triple Integrals

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Example Questions

Example Question #171 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{0}^{1}\int_{-4}^{0}\int_{0}^{1}(12z^3cos{(x)})dxdydz\end{align*}$ $\displaystyle \begin{align*}&\text{Evaluate the triple integral}\int_{0}^{1}\int_{-4}^{0}\int_{0}^{1}(12z^3cos{(x)})dxdydz\end{align*}$

Possible Answers:

$\displaystyle 3sin{(4)}$

$\displaystyle -768sin{(1)}$

$\displaystyle -12sin{(1)}$

$\displaystyle 12sin{(1)}$

$\displaystyle 768sin{(1)}$

Correct answer:

$\displaystyle 12sin{(1)}$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{0}^{1}\int_{-4}^{0}\int_{0}^{1}(12z^3cos{(x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{0}^{1}\int_{-4}^{0}\int_{0}^{1}(12z^3cos{(x)})dxdydz=\int_{0}^{1}\int_{-4}^{0}(12z^3sin{(x)})dydz|_{0}^{1}\\&\int_{0}^{1}\int_{-4}^{0}(12z^3sin{(1)})dydz=\int_{0}^{1}(12yz^3sin{(1)})dz|_{-4}^{0}\\&\int_{0}^{1}(48z^3sin{(1)})dz=12z^4sin{(1)}dz|_{0}^{1}=12sin{(1)}\end{align*}$ $\displaystyle \begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{0}^{1}\int_{-4}^{0}\int_{0}^{1}(12z^3cos{(x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{0}^{1}\int_{-4}^{0}\int_{0}^{1}(12z^3cos{(x)})dxdydz=\int_{0}^{1}\int_{-4}^{0}(12z^3sin{(x)})dydz|_{0}^{1}\\&\int_{0}^{1}\int_{-4}^{0}(12z^3sin{(1)})dydz=\int_{0}^{1}(12yz^3sin{(1)})dz|_{-4}^{0}\\&\int_{0}^{1}(48z^3sin{(1)})dz=12z^4sin{(1)}dz|_{0}^{1}=12sin{(1)}\end{align*}$

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Example Question #172 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{2}\int_{-3}^{1}\int_{-2}^{3}(9y^2sin{(5x)})dxdydz\end{align*}$ $\displaystyle \begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{2}\int_{-3}^{1}\int_{-2}^{3}(9y^2sin{(5x)})dxdydz\end{align*}$

Possible Answers:

$\displaystyle 84cos{(10)} - 84cos{(15)}$

$\displaystyle 105cos{(15)} - 105cos{(5)}$

$\displaystyle 84cos{(15)} + 84cos{(10)}$

$\displaystyle 84cos{(10)} + 84cos{(15)}$

$\displaystyle 84cos{(15)} - 84cos{(10)}$

Correct answer:

$\displaystyle 84cos{(10)} - 84cos{(15)}$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-3}^{2}\int_{-3}^{1}\int_{-2}^{3}(9y^2sin{(5x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{2}\int_{-3}^{1}\int_{-2}^{3}(9y^2sin{(5x)})dxdydz=\int_{-3}^{2}\int_{-3}^{1}(-{(9y^2cos{(5x)})}/5)dydz|_{-2}^{3}\\&\int_{-3}^{2}\int_{-3}^{1}(9y^2{(cos{(10)}/5 - cos{(15)}/5)})dydz=\int_{-3}^{2}(y^3{({(3cos{(10)})}/5 - {(3cos{(15)})}/5)})dz|_{-3}^{1}\\&\int_{-3}^{2}({(84cos{(10)})}/5 - {(84cos{(15)})}/5)dz=z{({(84cos{(10)})}/5 - {(84cos{(15)})}/5)}dz|_{-3}^{2}=84cos{(10)} - 84cos{(15)}\end{align*}$ $\displaystyle \begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-3}^{2}\int_{-3}^{1}\int_{-2}^{3}(9y^2sin{(5x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{2}\int_{-3}^{1}\int_{-2}^{3}(9y^2sin{(5x)})dxdydz=\int_{-3}^{2}\int_{-3}^{1}(-{(9y^2cos{(5x)})}/5)dydz|_{-2}^{3}\\&\int_{-3}^{2}\int_{-3}^{1}(9y^2{(cos{(10)}/5 - cos{(15)}/5)})dydz=\int_{-3}^{2}(y^3{({(3cos{(10)})}/5 - {(3cos{(15)})}/5)})dz|_{-3}^{1}\\&\int_{-3}^{2}({(84cos{(10)})}/5 - {(84cos{(15)})}/5)dz=z{({(84cos{(10)})}/5 - {(84cos{(15)})}/5)}dz|_{-3}^{2}=84cos{(10)} - 84cos{(15)}\end{align*}$

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Example Question #173 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{2}^{6}\int_{-1}^{3}\int_{2}^{5}(-6ysin{(x)})dxdydz\end{align*}$ $\displaystyle \begin{align*}&\text{Evaluate the triple integral}\int_{2}^{6}\int_{-1}^{3}\int_{2}^{5}(-6ysin{(x)})dxdydz\end{align*}$

Possible Answers:

$\displaystyle 72cos{(6)} - 72cos{(2)}$

$\displaystyle 384cos{(5)} - 384cos{(2)}$

$\displaystyle 252cos{(3)} - 252cos{(1)}$

$\displaystyle 252cos{(6)} - 252cos{(2)}$

$\displaystyle 96cos{(5)} - 96cos{(2)}$

Correct answer:

$\displaystyle 96cos{(5)} - 96cos{(2)}$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{6}\int_{-1}^{3}\int_{2}^{5}(-6ysin{(x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{6}\int_{-1}^{3}\int_{2}^{5}(-6ysin{(x)})dxdydz=\int_{2}^{6}\int_{-1}^{3}(6ycos{(x)})dydz|_{2}^{5}\\&\int_{2}^{6}\int_{-1}^{3}(-6y{(cos{(2)} - cos{(5)})})dydz=\int_{2}^{6}(-y^2{(3cos{(2)} - 3cos{(5)})})dz|_{-1}^{3}\\&\int_{2}^{6}(24cos{(5)} - 24cos{(2)})dz=-z{(24cos{(2)} - 24cos{(5)})}dz|_{2}^{6}=96cos{(5)} - 96cos{(2)}\end{align*}$ $\displaystyle \begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{6}\int_{-1}^{3}\int_{2}^{5}(-6ysin{(x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{6}\int_{-1}^{3}\int_{2}^{5}(-6ysin{(x)})dxdydz=\int_{2}^{6}\int_{-1}^{3}(6ycos{(x)})dydz|_{2}^{5}\\&\int_{2}^{6}\int_{-1}^{3}(-6y{(cos{(2)} - cos{(5)})})dydz=\int_{2}^{6}(-y^2{(3cos{(2)} - 3cos{(5)})})dz|_{-1}^{3}\\&\int_{2}^{6}(24cos{(5)} - 24cos{(2)})dz=-z{(24cos{(2)} - 24cos{(5)})}dz|_{2}^{6}=96cos{(5)} - 96cos{(2)}\end{align*}$

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Example Question #173 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{-2}\int_{-3}^{1}\int_{1}^{6}(4y - 4x^3)dxdydz\end{align*}$ $\displaystyle \begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{-2}\int_{-3}^{1}\int_{1}^{6}(4y - 4x^3)dxdydz\end{align*}$

Possible Answers:

680 $\displaystyle 680$

$\displaystyle -5260$

1220 $\displaystyle 1220$

1580 $\displaystyle 1580$

$\displaystyle -5380$

Correct answer:

$\displaystyle -5260$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-3}^{-2}\int_{-3}^{1}\int_{1}^{6}(4y - 4x^3)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{-2}\int_{-3}^{1}\int_{1}^{6}(4y - 4x^3)dxdydz=\int_{-3}^{-2}\int_{-3}^{1}(4xy - x^4)dydz|_{1}^{6}\\&\int_{-3}^{-2}\int_{-3}^{1}(20y - 1295)dydz=\int_{-3}^{-2}(5y{(2y - 259)})dz|_{-3}^{1}\\&\int_{-3}^{-2}(-5260)dz=-5260zdz|_{-3}^{-2}=-5260\end{align*}$ $\displaystyle \begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-3}^{-2}\int_{-3}^{1}\int_{1}^{6}(4y - 4x^3)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{-2}\int_{-3}^{1}\int_{1}^{6}(4y - 4x^3)dxdydz=\int_{-3}^{-2}\int_{-3}^{1}(4xy - x^4)dydz|_{1}^{6}\\&\int_{-3}^{-2}\int_{-3}^{1}(20y - 1295)dydz=\int_{-3}^{-2}(5y{(2y - 259)})dz|_{-3}^{1}\\&\int_{-3}^{-2}(-5260)dz=-5260zdz|_{-3}^{-2}=-5260\end{align*}$

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Example Question #174 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{5}\int_{0}^{4}\int_{1}^{4}(80x - 120z^2)dxdydz\end{align*}$ $\displaystyle \begin{align*}&\text{Evaluate the triple integral}\int_{3}^{5}\int_{0}^{4}\int_{1}^{4}(80x - 120z^2)dxdydz\end{align*}$

Possible Answers:

$\displaystyle -42240$

$\displaystyle -12480$

$\displaystyle -7680$

$\displaystyle -10560$

$\displaystyle -43200$

Correct answer:

$\displaystyle -42240$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{3}^{5}\int_{0}^{4}\int_{1}^{4}(80x - 120z^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{5}\int_{0}^{4}\int_{1}^{4}(80x - 120z^2)dxdydz=\int_{3}^{5}\int_{0}^{4}(40x{(x - 3z^2)})dydz|_{1}^{4}\\&\int_{3}^{5}\int_{0}^{4}(600 - 360z^2)dydz=\int_{3}^{5}(-y{(360z^2 - 600)})dz|_{0}^{4}\\&\int_{3}^{5}(2400 - 1440z^2)dz=-480z{(z^2 - 5)}dz|_{3}^{5}=-42240\end{align*}$ $\displaystyle \begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{3}^{5}\int_{0}^{4}\int_{1}^{4}(80x - 120z^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{5}\int_{0}^{4}\int_{1}^{4}(80x - 120z^2)dxdydz=\int_{3}^{5}\int_{0}^{4}(40x{(x - 3z^2)})dydz|_{1}^{4}\\&\int_{3}^{5}\int_{0}^{4}(600 - 360z^2)dydz=\int_{3}^{5}(-y{(360z^2 - 600)})dz|_{0}^{4}\\&\int_{3}^{5}(2400 - 1440z^2)dz=-480z{(z^2 - 5)}dz|_{3}^{5}=-42240\end{align*}$

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Example Question #175 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-4}^{1}\int_{5}^{10}\int_{-2}^{1}(6{(x + 6)}^2)dxdydz\end{align*}$ $\displaystyle \begin{align*}&\text{Evaluate the triple integral}\int_{-4}^{1}\int_{5}^{10}\int_{-2}^{1}(6{(x + 6)}^2)dxdydz\end{align*}$

Possible Answers:

10050 $\displaystyle 10050$

13950 $\displaystyle 13950$

24750 $\displaystyle 24750$

4050 $\displaystyle 4050$

82950 $\displaystyle 82950$

Correct answer:

13950 $\displaystyle 13950$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-4}^{1}\int_{5}^{10}\int_{-2}^{1}(6{(x + 6)}^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4}^{1}\int_{5}^{10}\int_{-2}^{1}(6{(x + 6)}^2)dxdydz=\int_{-4}^{1}\int_{5}^{10}(2{(x + 6)}^3)dydz|_{-2}^{1}\\&\int_{-4}^{1}\int_{5}^{10}(558)dydz=\int_{-4}^{1}(558y)dz|_{5}^{10}\\&\int_{-4}^{1}(2790)dz=2790zdz|_{-4}^{1}=13950\end{align*}$ $\displaystyle \begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-4}^{1}\int_{5}^{10}\int_{-2}^{1}(6{(x + 6)}^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4}^{1}\int_{5}^{10}\int_{-2}^{1}(6{(x + 6)}^2)dxdydz=\int_{-4}^{1}\int_{5}^{10}(2{(x + 6)}^3)dydz|_{-2}^{1}\\&\int_{-4}^{1}\int_{5}^{10}(558)dydz=\int_{-4}^{1}(558y)dz|_{5}^{10}\\&\int_{-4}^{1}(2790)dz=2790zdz|_{-4}^{1}=13950\end{align*}$

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Example Question #171 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{4}\int_{3}^{4}\int_{5}^{8}(4{(y - 2)}^3)dxdydz\end{align*}$ $\displaystyle \begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{4}\int_{3}^{4}\int_{5}^{8}(4{(y - 2)}^3)dxdydz\end{align*}$

Possible Answers:

3240 $\displaystyle 3240$

6075 $\displaystyle 6075$

$\displaystyle -195$

225 $\displaystyle 225$

8050 $\displaystyle 8050$

Correct answer:

225 $\displaystyle 225$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{4}\int_{3}^{4}\int_{5}^{8}(4{(y - 2)}^3)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{4}\int_{3}^{4}\int_{5}^{8}(4{(y - 2)}^3)dxdydz=\int_{-1}^{4}\int_{3}^{4}(4x{(y - 2)}^3)dydz|_{5}^{8}\\&\int_{-1}^{4}\int_{3}^{4}(12{(y - 2)}^3)dydz=\int_{-1}^{4}(3{(y - 2)}^4)dz|_{3}^{4}\\&\int_{-1}^{4}(45)dz=45zdz|_{-1}^{4}=225\end{align*}$ $\displaystyle \begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{4}\int_{3}^{4}\int_{5}^{8}(4{(y - 2)}^3)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{4}\int_{3}^{4}\int_{5}^{8}(4{(y - 2)}^3)dxdydz=\int_{-1}^{4}\int_{3}^{4}(4x{(y - 2)}^3)dydz|_{5}^{8}\\&\int_{-1}^{4}\int_{3}^{4}(12{(y - 2)}^3)dydz=\int_{-1}^{4}(3{(y - 2)}^4)dz|_{3}^{4}\\&\int_{-1}^{4}(45)dz=45zdz|_{-1}^{4}=225\end{align*}$

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Example Question #44 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{9}\int_{2}^{3}\int_{3}^{8}(4{(y - 7)}^3 - 4x)dxdydz\end{align*}$ $\displaystyle \begin{align*}&\text{Evaluate the triple integral}\int_{4}^{9}\int_{2}^{3}\int_{3}^{8}(4{(y - 7)}^3 - 4x)dxdydz\end{align*}$

Possible Answers:

$\displaystyle -9775$

$\displaystyle -9875$

$\displaystyle -875$

$\displaystyle -1925$

$\displaystyle -1525$

Correct answer:

$\displaystyle -9775$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{9}\int_{2}^{3}\int_{3}^{8}(4{(y - 7)}^3 - 4x)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{9}\int_{2}^{3}\int_{3}^{8}(4{(y - 7)}^3 - 4x)dxdydz=\int_{4}^{9}\int_{2}^{3}(4x{(y - 7)}^3 - 2x^2)dydz|_{3}^{8}\\&\int_{4}^{9}\int_{2}^{3}(20{(y - 7)}^3 - 110)dydz=\int_{4}^{9}(5{({(y - 7)}^3 - 22)}{(y - 7)})dz|_{2}^{3}\\&\int_{4}^{9}(-1955)dz=-1955zdz|_{4}^{9}=-9775\end{align*}$ $\displaystyle \begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{9}\int_{2}^{3}\int_{3}^{8}(4{(y - 7)}^3 - 4x)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{9}\int_{2}^{3}\int_{3}^{8}(4{(y - 7)}^3 - 4x)dxdydz=\int_{4}^{9}\int_{2}^{3}(4x{(y - 7)}^3 - 2x^2)dydz|_{3}^{8}\\&\int_{4}^{9}\int_{2}^{3}(20{(y - 7)}^3 - 110)dydz=\int_{4}^{9}(5{({(y - 7)}^3 - 22)}{(y - 7)})dz|_{2}^{3}\\&\int_{4}^{9}(-1955)dz=-1955zdz|_{4}^{9}=-9775\end{align*}$

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Example Question #172 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{2}\int_{5}^{6}\int_{2}^{4}(2x - 4z)dxdydz\end{align*}$ $\displaystyle \begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{2}\int_{5}^{6}\int_{2}^{4}(2x - 4z)dxdydz\end{align*}$

Possible Answers:

$\displaystyle -66$

$\displaystyle -96$

$\displaystyle 24$

$\displaystyle 54$

$\displaystyle -126$

Correct answer:

$\displaystyle 24$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{2}\int_{5}^{6}\int_{2}^{4}(2x - 4z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{2}\int_{5}^{6}\int_{2}^{4}(2x - 4z)dxdydz=\int_{-1}^{2}\int_{5}^{6}(x{(x - 4z)})dydz|_{2}^{4}\\&\int_{-1}^{2}\int_{5}^{6}(12 - 8z)dydz=\int_{-1}^{2}(-y{(8z - 12)})dz|_{5}^{6}\\&\int_{-1}^{2}(12 - 8z)dz=-4z{(z - 3)}dz|_{-1}^{2}=24\end{align*}$ $\displaystyle \begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{2}\int_{5}^{6}\int_{2}^{4}(2x - 4z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{2}\int_{5}^{6}\int_{2}^{4}(2x - 4z)dxdydz=\int_{-1}^{2}\int_{5}^{6}(x{(x - 4z)})dydz|_{2}^{4}\\&\int_{-1}^{2}\int_{5}^{6}(12 - 8z)dydz=\int_{-1}^{2}(-y{(8z - 12)})dz|_{5}^{6}\\&\int_{-1}^{2}(12 - 8z)dz=-4z{(z - 3)}dz|_{-1}^{2}=24\end{align*}$

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Example Question #176 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{3}\int_{-3}^{0}\int_{3}^{7}(y^3 - x)dxdydz\end{align*}$ $\displaystyle \begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{3}\int_{-3}^{0}\int_{3}^{7}(y^3 - x)dxdydz\end{align*}$

Possible Answers:

$\displaystyle -564$

6912 $\displaystyle 6912$

7032 $\displaystyle 7032$

$\displaystyle 0$

$\displaystyle -372$

Correct answer:

$\displaystyle -564$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{3}\int_{-3}^{0}\int_{3}^{7}(y^3 - x)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{3}\int_{-3}^{0}\int_{3}^{7}(y^3 - x)dxdydz=\int_{-1}^{3}\int_{-3}^{0}(-{(x{(x - 2y^3)})}/2)dydz|_{3}^{7}\\&\int_{-1}^{3}\int_{-3}^{0}(4y^3 - 20)dydz=\int_{-1}^{3}(y^4 - 20y)dz|_{-3}^{0}\\&\int_{-1}^{3}(-141)dz=-141zdz|_{-1}^{3}=-564\end{align*}$ $\displaystyle \begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{3}\int_{-3}^{0}\int_{3}^{7}(y^3 - x)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{3}\int_{-3}^{0}\int_{3}^{7}(y^3 - x)dxdydz=\int_{-1}^{3}\int_{-3}^{0}(-{(x{(x - 2y^3)})}/2)dydz|_{3}^{7}\\&\int_{-1}^{3}\int_{-3}^{0}(4y^3 - 20)dydz=\int_{-1}^{3}(y^4 - 20y)dz|_{-3}^{0}\\&\int_{-1}^{3}(-141)dz=-141zdz|_{-1}^{3}=-564\end{align*}$

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Calculus 3 : Triple Integrals

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