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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #201 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-4}^{-2}\int_{-10}^{-7}\int_{-6}^{-4.5}(\frac{(12cos(3y)sin(3x)e^{(-z)})}{79})dxdydz\end{align*}$

Possible Answers:

-0.1

0.29

0.03

-0.02

Correct answer:

-0.1

Explanation:

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

$\begin{align*}&\int_{-4}^{-2}\int_{-10}^{-7}\int_{-6}^{-4.5}(\frac{(12cos(3y)sin(3x)e^{(-z)})}{79})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4}^{-2}\int_{-10}^{-7}\int_{-6}^{-4.5}(\frac{(12cos(3y)sin(3x)e^{(-z)})}{79})dxdydz=\int_{-4}^{-2}\int_{-10}^{-7}(-\frac{(4cos(3x)cos(3y)e^{(-z)})}{79})dydz|_{-6}^{-4.5}\\&\int_{-4}^{-2}\int_{-10}^{-7}(\frac{(4cos(3y)e^{(-z)}\cdot (cos(18) - cos(\frac{27}{2})))}{79})dydz=\int_{-4}^{-2}(\frac{(4sin(3y)e^{(-z)}\cdot (cos(18) - cos(\frac{27}{2})))}{237})dz|_{-10}^{-7}\\&\int_{-4}^{-2}(-\frac{(4e^{(-z)}\cdot (cos(18) - cos(\frac{27}{2}))\cdot (sin(21) - sin(30)))}{237})dz=\frac{(4e^{(-z)}\cdot (cos(18) - cos(\frac{27}{2}))\cdot (sin(21) - sin(30)))}{237}|_{-4}^{-2}=-0.1\end{align*}$

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Example Question #331 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{3.5}^{8.5}\int_{8}^{13}\int_{4.5}^{7}(\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-x)})}{(8y)})dxdydz\end{align*}$

Possible Answers:

0.36

-0.09

0.73

-0.73

Correct answer:

0.36

Explanation:

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

$\begin{align*}&\int_{3.5}^{8.5}\int_{8}^{13}\int_{4.5}^{7}(\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-x)})}{(8y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3.5}^{8.5}\int_{8}^{13}\int_{4.5}^{7}(\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-x)})}{(8y)})dxdydz=\int_{3.5}^{8.5}\int_{8}^{13}(-\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-x)})}{(8y)})dydz|_{4.5}^{7}\\&\int_{3.5}^{8.5}\int_{8}^{13}(\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-7)}\cdot (e^{(\frac{5}{2})} - 1))}{(8y)})dydz=\int_{3.5}^{8.5}(\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-7)}ln(y)\cdot (e^{(\frac{5}{2})} - 1))}{8})dz|_{8}^{13}\\&\int_{3.5}^{8.5}(\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-7)}ln(\frac{13}{8})\cdot (e^{(\frac{5}{2})} - 1))}{8})dz=\frac{(13\cdot 2^{(\frac{z}{2})}e^{(-7)}ln(\frac{13}{8})\cdot (e^{(\frac{5}{2})} - 1))}{(4ln(2))}|_{3.5}^{8.5}=0.36\end{align*}$

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Example Question #332 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-2.5}\int_{-5}^{2.5}\int_{6}^{7.5}(\frac{(3^{(\frac{y}{2})}cos(3z))}{(18x)})dxdydz\end{align*}$

Possible Answers:

0.1

0.02

-0.15

-0.05

Correct answer:

-0.05

Explanation:

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

$\begin{align*}&\int_{-6}^{-2.5}\int_{-5}^{2.5}\int_{6}^{7.5}(\frac{(3^{(\frac{y}{2})}cos(3z))}{(18x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-2.5}\int_{-5}^{2.5}\int_{6}^{7.5}(\frac{(3^{(\frac{y}{2})}cos(3z))}{(18x)})dxdydz=\int_{-6}^{-2.5}\int_{-5}^{2.5}(\frac{(3^{(\frac{y}{2})}cos(3z)ln(x))}{18})dydz|_{6}^{7.5}\\&\int_{-6}^{-2.5}\int_{-5}^{2.5}(\frac{(3^{(\frac{y}{2})}cos(3z)ln(\frac{5}{4}))}{18})dydz=\int_{-6}^{-2.5}(\frac{(3^{(\frac{y}{2})}cos(3z)ln(\frac{5}{4}))}{(9ln(3))})dz|_{-5}^{2.5}\\&\int_{-6}^{-2.5}(-\frac{(cos(3z)ln(\frac{5}{4})\cdot (2\cdot 3^{(\frac{1}{2})} - 162\cdot 3^{(\frac{1}{4})}))}{(486ln(3))})dz=-\frac{(sin(3z)ln(\frac{5}{4})\cdot (2\cdot 3^{(\frac{1}{2})} - 162\cdot 3^{(\frac{1}{4})}))}{(1458ln(3))}|_{-6}^{-2.5}=-0.05\end{align*}$

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Example Question #333 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{11}\int_{10}^{12.5}\int_{4.5}^{9}(\frac{(54\cdot 3^{(\frac{x}{3})}cos(3z)sin(3y))}{11})dxdydz\end{align*}$

Possible Answers:

187.9

23.48

-281.8

-46.97

Correct answer:

-46.97

Explanation:

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

$\begin{align*}&\int_{6}^{11}\int_{10}^{12.5}\int_{4.5}^{9}(\frac{(54\cdot 3^{(\frac{x}{3})}cos(3z)sin(3y))}{11})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{11}\int_{10}^{12.5}\int_{4.5}^{9}(\frac{(54\cdot 3^{(\frac{x}{3})}cos(3z)sin(3y))}{11})dxdydz=\int_{6}^{11}\int_{10}^{12.5}(\frac{(162\cdot 3^{(\frac{x}{3})}cos(3z)sin(3y))}{(11ln(3))})dydz|_{4.5}^{9}\\&\int_{6}^{11}\int_{10}^{12.5}(-\frac{(486cos(3z)sin(3y)\cdot (3^{(\frac{1}{2})} - 9))}{(11ln(3))})dydz=\int_{6}^{11}(\frac{(162cos(3y)cos(3z)\cdot (3^{(\frac{1}{2})} - 9))}{(11ln(3))})dz|_{10}^{12.5}\\&\int_{6}^{11}(-\frac{(162cos(3z)\cdot (cos(30) - cos(\frac{75}{2}))\cdot (3^{(\frac{1}{2})} - 9))}{(11ln(3))})dz=-\frac{(54sin(3z)\cdot (cos(30) - cos(\frac{75}{2}))\cdot (3^{(\frac{1}{2})} - 9))}{(11ln(3))}|_{6}^{11}=-46.97\end{align*}$

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Example Question #331 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-5.5}\int_{-10}^{-9}\int_{-4.5}^{0.5}(\frac{(5sin(y + 2)e^{(2x)})}{(6z^{2})})dxdydz\end{align*}$

Possible Answers:

0.02

0.07

-0.07

-0.22

Correct answer:

-0.07

Explanation:

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

$\begin{align*}&\int_{-9}^{-5.5}\int_{-10}^{-9}\int_{-4.5}^{0.5}(\frac{(5sin(y + 2)e^{(2x)})}{(6z^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-5.5}\int_{-10}^{-9}\int_{-4.5}^{0.5}(\frac{(5sin(y + 2)e^{(2x)})}{(6z^{2})})dxdydz=\int_{-9}^{-5.5}\int_{-10}^{-9}(\frac{(5sin(y + 2)e^{(2x)})}{(12z^{2})})dydz|_{-4.5}^{0.5}\\&\int_{-9}^{-5.5}\int_{-10}^{-9}(\frac{(5sin(y + 2)e^{(-9)}\cdot (e^{(10)} - 1))}{(12z^{2})})dydz=\int_{-9}^{-5.5}(-\frac{(5cos(y + 2)e^{(-9)}\cdot (e^{(10)} - 1))}{(12z^{2})})dz|_{-10}^{-9}\\&\int_{-9}^{-5.5}(-\frac{(5e^{(-9)}\cdot (cos(7) - cos(8))\cdot (e^{(10)} - 1))}{(12z^{2})})dz=\frac{(5e^{(-9)}\cdot (cos(7) - cos(8))\cdot (e^{(10)} - 1))}{(12z)}|_{-9}^{-5.5}=-0.07\end{align*}$

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Example Question #334 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-5}^{-1.5}\int_{6}^{9.5}\int_{-8}^{-5.5}(8\cdot 2^{(\frac{z}{2})}cos(x + 1)cos(4y))dxdydz\end{align*}$

Possible Answers:

1.58

-9.47

4.74

-0.95

Correct answer:

4.74

Explanation:

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

$\begin{align*}&\int_{-5}^{-1.5}\int_{6}^{9.5}\int_{-8}^{-5.5}(8\cdot 2^{(\frac{z}{2})}cos(x + 1)cos(4y))dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-5}^{-1.5}\int_{6}^{9.5}\int_{-8}^{-5.5}(8\cdot 2^{(\frac{z}{2})}cos(x + 1)cos(4y))dxdydz=\int_{-5}^{-1.5}\int_{6}^{9.5}(8\cdot 2^{(\frac{z}{2})}cos(4y)sin(x + 1))dydz|_{-8}^{-5.5}\\&\int_{-5}^{-1.5}\int_{6}^{9.5}(8\cdot 2^{(\frac{z}{2})}cos(4y)\cdot (sin(7) - sin(\frac{9}{2})))dydz=\int_{-5}^{-1.5}(2\cdot 2^{(\frac{z}{2})}sin(4y)\cdot (sin(7) - sin(\frac{9}{2})))dz|_{6}^{9.5}\\&\int_{-5}^{-1.5}(-2\cdot 2^{(\frac{z}{2})}\cdot (sin(7) - sin(\frac{9}{2}))\cdot (sin(24) - sin(38)))dz=-\frac{(4\cdot 2^{(\frac{z}{2})}\cdot (\frac{cos(17)}{2}- cos(31) -\frac{ cos(\frac{39}{2})}{2}+\frac{ cos(45)}{2}+\frac{ cos(\frac{57}{2})}{2}+\frac{ cos(\frac{67}{2})}{2}-\frac{ cos(\frac{85}{2})}{2}))}{ln(2)}|_{-5}^{-1.5}=4.74\end{align*}$

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Example Question #335 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{5.5}\int_{-3.5}^{1.5}\int_{5}^{7.5}(\frac{(e^{(2y)}e^{(z)})}{(5x)})dxdydz\end{align*}$

Possible Answers:

-45.73

-182.9

182.9

1097

Correct answer:

182.9

Explanation:

$\begin{align*}&\int_{3}^{5.5}\int_{-3.5}^{1.5}\int_{5}^{7.5}(\frac{(e^{(2y)}e^{(z)})}{(5x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{5.5}\int_{-3.5}^{1.5}\int_{5}^{7.5}(\frac{(e^{(2y)}e^{(z)})}{(5x)})dxdydz=\int_{3}^{5.5}\int_{-3.5}^{1.5}(\frac{(e^{(2y + z)}ln(x))}{5})dydz|_{5}^{7.5}\\&\int_{3}^{5.5}\int_{-3.5}^{1.5}(\frac{(e^{(2y + z)}ln(\frac{3}{2}))}{5})dydz=\int_{3}^{5.5}(\frac{(e^{(2y + z)}ln(\frac{3}{2}))}{10})dz|_{-3.5}^{1.5}\\&\int_{3}^{5.5}(\frac{(e^{(-7)}e^{(z)}ln(\frac{3}{2})\cdot (e^{(10)} - 1))}{10})dz=\frac{(e^{(z - 7)}ln(\frac{3}{2})\cdot (e^{(10)} - 1))}{10}|_{3}^{5.5}=182.9\end{align*}$

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Example Question #341 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{4.5}^{6}\int_{3.5}^{4.5}\int_{-4}^{1}(\frac{(33\cdot 2^ze^{(2x)}e^{(-y)})}{4})dxdydz\end{align*}$

Possible Answers:

34.73

8.68

-69.45

-8.68

Correct answer:

34.73

Explanation:

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

$\begin{align*}&\int_{4.5}^{6}\int_{3.5}^{4.5}\int_{-4}^{1}(\frac{(33\cdot 2^ze^{(2x)}e^{(-y)})}{4})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4.5}^{6}\int_{3.5}^{4.5}\int_{-4}^{1}(\frac{(33\cdot 2^ze^{(2x)}e^{(-y)})}{4})dxdydz=\int_{4.5}^{6}\int_{3.5}^{4.5}(\frac{(33\cdot 2^ze^{(2x - y)})}{8})dydz|_{-4}^{1}\\&\int_{4.5}^{6}\int_{3.5}^{4.5}(\frac{(33\cdot 2^ze^{(-8)}e^{(-y)}\cdot (e^{(10)} - 1))}{8})dydz=\int_{4.5}^{6}(-\frac{(33\cdot 2^ze^{(-8)}e^{(-y)}\cdot (e^{(10)} - 1))}{8})dz|_{3.5}^{4.5}\\&\int_{4.5}^{6}(\frac{(33\cdot 2^z\cdot (e^{(-\frac{5}{2})} - e^{(-\frac{25}{2})})\cdot (e^{(1)} - 1))}{8})dz=\frac{(33\cdot 2^ze^{(-\frac{25}{2})}\cdot (e^{(1)} - 1)\cdot (e^{(10)} - 1))}{(8ln(2))}|_{4.5}^{6}=34.73\end{align*}$

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Example Question #342 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-10}^{-6}\int_{10}^{12.5}\int_{-3.5}^{1.75}(\frac{(29\cdot 2^{(\frac{x}{4})}sin(y + 1))}{z^{2}})dxdydz\end{align*}$

Possible Answers:

-5.33

-0.89

0.89

31.98

Correct answer:

-5.33

Explanation:

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

$\begin{align*}&\int_{-10}^{-6}\int_{10}^{12.5}\int_{-3.5}^{1.75}(\frac{(29\cdot 2^{(\frac{x}{4})}sin(y + 1))}{z^{2}})dxdydz\\&\text{The approach is simply to take it step by step:}\end{align*}$

$\begin{align*}\\&\int_{-10}^{-6}\int_{10}^{12.5}\int_{-3.5}^{1.75}(\frac{(29\cdot 2^{(\frac{x}{4})}sin(y + 1))}{z^{2}})dxdydz=\int_{-10}^{-6}\int_{10}^{12.5}(\frac{(116\cdot 2^{(\frac{x}{4})}sin(y + 1))}{(z^{2}ln(2))})dydz|_{-3.5}^{1.75}\\&\int_{-10}^{-6}\int_{10}^{12.5}(-\frac{(29sin(y + 1)\cdot (2\cdot 2^{(\frac{1}{8})} - 4\cdot 2^{(\frac{7}{16})}))}{(z^{2}ln(2))})dydz=\int_{-10}^{-6}(\frac{(29cos(y + 1)\cdot (2\cdot 2^{(\frac{1}{8})} - 4\cdot 2^{(\frac{7}{16})}))}{(z^{2}ln(2))})dz|_{10}^{12.5}\\&\int_{-10}^{-6}(-\frac{(58\cdot 2^{(\frac{1}{8})}cos(11) - 116\cdot 2^{(\frac{7}{16})}cos(11) - 58\cdot 2^{(\frac{1}{8})}cos(\frac{27}{2}) + 116\cdot 2^{(\frac{7}{16})}cos(\frac{27}{2}))}{(z^{2}ln(2))})dz=\frac{(58\cdot 2^{(\frac{1}{8})}cos(11) - 58\cdot 2^{(\frac{1}{8})}cos(\frac{27}{2}) - 116\cdot 128^{(\frac{1}{16})}cos(11) + 116\cdot 128^{(\frac{1}{16})}cos(\frac{27}{2}))}{(zln(2))}|_{-10}^{-6}=-5.33\end{align*}$

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Example Question #341 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3.5}^{1.5}\int_{-9}^{-7.5}\int_{9}^{10}(\frac{(7sin(4y)e^{(z)})}{65})dxdydz\end{align*}$

Possible Answers:

-0.17

-0.03

0.01

0.14

Correct answer:

-0.03

Explanation:

$\begin{align*}&\int_{-3.5}^{1.5}\int_{-9}^{-7.5}\int_{9}^{10}(\frac{(7sin(4y)e^{(z)})}{65})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3.5}^{1.5}\int_{-9}^{-7.5}\int_{9}^{10}(\frac{(7sin(4y)e^{(z)})}{65})dxdydz=\int_{-3.5}^{1.5}\int_{-9}^{-7.5}(\frac{(7xsin(4y)e^{(z)})}{65})dydz|_{9}^{10}\\&\int_{-3.5}^{1.5}\int_{-9}^{-7.5}(\frac{(7sin(4y)e^{(z)})}{65})dydz=\int_{-3.5}^{1.5}(-\frac{(7cos(4y)e^{(z)})}{260})dz|_{-9}^{-7.5}\\&\int_{-3.5}^{1.5}(-\frac{(7e^{(z)}\cdot (cos(30) - cos(36)))}{260})dz=-e^{(z)}\cdot (\frac{(7cos(30))}{260}-\frac{ (7cos(36))}{260})|_{-3.5}^{1.5}=-0.03\end{align*}$

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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #201 : Triple Integration In Cartesian Coordinates

Example Question #331 : Triple Integrals

Example Question #332 : Triple Integrals

Example Question #333 : Triple Integrals

Example Question #331 : Multiple Integration

Example Question #334 : Triple Integrals

Example Question #335 : Triple Integrals

Example Question #341 : Multiple Integration

Example Question #342 : Triple Integrals

Example Question #341 : Triple Integrals

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