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Calculus 3 : Triple Integration in Cartesian Coordinates

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Example Questions

Example Question #324 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-7.5}\int_{4.5}^{8}\int_{8}^{12.5}(\frac{(6\cdot3^ycos(z + 2)sin(4x))}{49})dxdydz\end{align*}$

Possible Answers:

-159.4

-31.87

7.97

127.5

Correct answer:

-31.87

Explanation:

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

$\begin{align*}&\int_{-9}^{-7.5}\int_{4.5}^{8}\int_{8}^{12.5}(\frac{(6\cdot3^ycos(z + 2)sin(4x))}{49})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-7.5}\int_{4.5}^{8}\int_{8}^{12.5}(\frac{(6\cdot3^ycos(z + 2)sin(4x))}{49})dxdydz=\int_{-9}^{-7.5}\int_{4.5}^{8}(-\frac{(3\cdot3^ycos(4x)cos(z + 2))}{98})dydz|_{8}^{12.5}\\&\int_{-9}^{-7.5}\int_{4.5}^{8}(\frac{(6\cdot3^ycos(z + 2)\cdot(\frac{cos(32)}{4}-\frac{ cos(50)}{4}))}{49})dydz=\int_{-9}^{-7.5}(\frac{(3\cdot3^y\cdot(cos(z + 2)cos(32) - cos(z + 2)cos(50)))}{(98ln(3))})dz|_{4.5}^{8}\\&\int_{-9}^{-7.5}(-\frac{(243cos(z + 2)\cdot(cos(32) - cos(50))\cdot(3^{(\frac{1}{2})} - 81))}{(98ln(3))})dz=-\frac{(243sin(z + 2)\cdot(cos(32) - cos(50))\cdot(3^{(\frac{1}{2})} - 81))}{(98ln(3))}|_{-9}^{-7.5}=-31.87\end{align*}$

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Example Question #325 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{11}\int_{-8}^{-5.5}\int_{-5}^{-4}(\frac{(13z^{2}e^{(-2x)})}{(7y)})dxdydz\end{align*}$

Possible Answers:

$-1.309\cdot10^{7}$

436500

$2.182\cdot10^{6}$

$-2.182\cdot10^{6}$

Correct answer:

$-2.182\cdot10^{6}$

Explanation:

$\begin{align*}&\int_{7}^{11}\int_{-8}^{-5.5}\int_{-5}^{-4}(\frac{(13z^{2}e^{(-2x)})}{(7y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{11}\int_{-8}^{-5.5}\int_{-5}^{-4}(\frac{(13z^{2}e^{(-2x)})}{(7y)})dxdydz=\int_{7}^{11}\int_{-8}^{-5.5}(-\frac{(13z^{2}e^{(-2x)})}{(14y)})dydz|_{-5}^{-4}\\&\int_{7}^{11}\int_{-8}^{-5.5}(\frac{(13z^{2}e^{(8)}\cdot(e^{(2)} - 1))}{(14y)})dydz=\int_{7}^{11}(\frac{(13z^{2}e^{(8)}ln(y)\cdot(e^{(2)} - 1))}{14})dz|_{-8}^{-5.5}\\&\int_{7}^{11}(\frac{(13z^{2}e^{(8)}ln(\frac{11}{16})\cdot(e^{(2)} - 1))}{14})dz=\frac{(13z^{3}e^{(8)}ln(\frac{11}{16})\cdot(e^{(2)} - 1))}{42}|_{7}^{11}=-2.182\cdot10^{6}\end{align*}$

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Example Question #326 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{9}\int_{-5}^{-2.5}\int_{8}^{11.5}(\frac{(9xe^{(y)})}{(11z)})dxdydz\end{align*}$

Possible Answers:

-0.11

-0.53

0.53

1.06

Correct answer:

0.53

Explanation:

$\begin{align*}&\int_{7}^{9}\int_{-5}^{-2.5}\int_{8}^{11.5}(\frac{(9xe^{(y)})}{(11z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{9}\int_{-5}^{-2.5}\int_{8}^{11.5}(\frac{(9xe^{(y)})}{(11z)})dxdydz=\int_{7}^{9}\int_{-5}^{-2.5}(\frac{(9x^{2}e^{(y)})}{(22z)})dydz|_{8}^{11.5}\\&\int_{7}^{9}\int_{-5}^{-2.5}(\frac{(2457e^{(y)})}{(88z)})dydz=\int_{7}^{9}(\frac{(2457e^{(y)})}{(88z)})dz|_{-5}^{-2.5}\\&\int_{7}^{9}(\frac{(2457e^{(-5)}\cdot(e^{(\frac{5}{2})} - 1))}{(88z)})dz=\frac{(2457e^{(-5)}ln(z)\cdot(e^{(\frac{5}{2})} - 1))}{88}|_{7}^{9}=0.53\end{align*}$

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Example Question #327 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{5.5}\int_{-10}^{-5.5}\int_{6}^{9.5}(27cos(3y)sin(4x)e^{(2z)})dxdydz\end{align*}$

Possible Answers:

9387

46933

Correct answer:

9387

Explanation:

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

$\begin{align*}&\int_{4}^{5.5}\int_{-10}^{-5.5}\int_{6}^{9.5}(27cos(3y)sin(4x)e^{(2z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{5.5}\int_{-10}^{-5.5}\int_{6}^{9.5}(27cos(3y)sin(4x)e^{(2z)})dxdydz=\int_{4}^{5.5}\int_{-10}^{-5.5}(-\frac{(27cos(4x)cos(3y)e^{(2z)})}{4})dydz|_{6}^{9.5}\\&\int_{4}^{5.5}\int_{-10}^{-5.5}(\frac{(27cos(3y)e^{(2z)}\cdot(cos(24) - cos(38)))}{4})dydz=\int_{4}^{5.5}(\frac{(9sin(3y)e^{(2z)}\cdot(cos(24) - cos(38)))}{4})dz|_{-10}^{-5.5}\\&\int_{4}^{5.5}(\frac{(9e^{(2z)}\cdot(cos(24) - cos(38))\cdot(sin(30) - sin(\frac{33}{2})))}{4})dz=\frac{(9e^{(2z)}\cdot(cos(24) - cos(38))\cdot(sin(30) - sin(\frac{33}{2})))}{8}|_{4}^{5.5}=9387\end{align*}$

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Example Question #328 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{8}^{11}\int_{9}^{13}\int_{-10}^{-6.5}(\frac{(3y^{2})}{2})dxdydz\end{align*}$

Possible Answers:

1927

7707

Correct answer:

7707

Explanation:

$\begin{align*}&\int_{8}^{11}\int_{9}^{13}\int_{-10}^{-6.5}(\frac{(3y^{2})}{2})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{8}^{11}\int_{9}^{13}\int_{-10}^{-6.5}(\frac{(3y^{2})}{2})dxdydz=\int_{8}^{11}\int_{9}^{13}(\frac{(3xy^{2})}{2})dydz|_{-10}^{-6.5}\\&\int_{8}^{11}\int_{9}^{13}(\frac{(21y^{2})}{4})dydz=\int_{8}^{11}(\frac{(7y^{3})}{4})dz|_{9}^{13}\\&\int_{8}^{11}(2569)dz=2569z|_{8}^{11}=7707\end{align*}$

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Example Question #329 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{10}\int_{7}^{8}\int_{-5}^{-3.5}(\frac{(17y^{2})}{(21\cdot2^xz)})dxdydz\end{align*}$

Possible Answers:

115.9

-115.9

695.2

Correct answer:

695.2

Explanation:

$\begin{align*}&\int_{6}^{10}\int_{7}^{8}\int_{-5}^{-3.5}(\frac{(17y^{2})}{(21\cdot2^xz)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{10}\int_{7}^{8}\int_{-5}^{-3.5}(\frac{(17y^{2})}{(21\cdot2^xz)})dxdydz=\int_{6}^{10}\int_{7}^{8}(-\frac{(17y^{2})}{(21\cdot2^xzln(2))})dydz|_{-5}^{-3.5}\\&\int_{6}^{10}\int_{7}^{8}(-\frac{(136y^{2}\cdot(2^{(\frac{1}{2})} - 4))}{(21zln(2))})dydz=\int_{6}^{10}(-\frac{(y^{3}\cdot(136\cdot2^{(\frac{1}{2})} - 544))}{(63zln(2))})dz|_{7}^{8}\\&\int_{6}^{10}(-\frac{(22984\cdot(2^{(\frac{1}{2})} - 4))}{(63zln(2))})dz=-\frac{(ln(z)\cdot(22984\cdot2^{(\frac{1}{2})} - 91936))}{(63ln(2))}|_{6}^{10}=695.2\end{align*}$

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Example Question #330 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{10.5}\int_{8}^{9}\int_{-4.5}^{-2.5}(\frac{1}{(9\cdot3^xz^{3})})dxdydz\end{align*}$

Possible Answers:

-0.02

-0.24

0.47

0.12

Correct answer:

0.12

Explanation:

$\begin{align*}&\int_{6}^{10.5}\int_{8}^{9}\int_{-4.5}^{-2.5}(\frac{1}{(9\cdot3^xz^{3})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{10.5}\int_{8}^{9}\int_{-4.5}^{-2.5}(\frac{1}{(9\cdot3^xz^{3})})dxdydz=\int_{6}^{10.5}\int_{8}^{9}(-\frac{1}{(9\cdot3^xz^{3}ln(3))})dydz|_{-4.5}^{-2.5}\\&\int_{6}^{10.5}\int_{8}^{9}(\frac{(8\cdot3^{(\frac{1}{2})})}{(z^{3}ln(3))})dydz=\int_{6}^{10.5}(\frac{(8\cdot3^{(\frac{1}{2})}y)}{(z^{3}ln(3))})dz|_{8}^{9}\\&\int_{6}^{10.5}(\frac{(8\cdot3^{(\frac{1}{2})})}{(z^{3}ln(3))})dz=-\frac{(4\cdot3^{(\frac{1}{2})})}{(z^{2}ln(3))}|_{6}^{10.5}=0.12\end{align*}$

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Example Question #331 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-4}\int_{2}^{3.5}\int_{3.5}^{5}(\frac{(9cos(4z)e^{(2x)})}{(73y)})dxdydz\end{align*}$

Possible Answers:

669

55.75

-111.5

Correct answer:

-111.5

Explanation:

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

$\begin{align*}&\int_{-6}^{-4}\int_{2}^{3.5}\int_{3.5}^{5}(\frac{(9cos(4z)e^{(2x)})}{(73y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-4}\int_{2}^{3.5}\int_{3.5}^{5}(\frac{(9cos(4z)e^{(2x)})}{(73y)})dxdydz=\int_{-6}^{-4}\int_{2}^{3.5}(\frac{(9cos(4z)e^{(2x)})}{(146y)})dydz|_{3.5}^{5}\\&\int_{-6}^{-4}\int_{2}^{3.5}(\frac{(9cos(4z)e^{(7)}\cdot(e^{(3)} - 1))}{(146y)})dydz=\int_{-6}^{-4}(-ln(y)\cdot(\frac{(9cos(4z)e^{(7)})}{146}-\frac{ (9cos(4z)e^{(10)})}{146}))dz|_{2}^{3.5}\\&\int_{-6}^{-4}(\frac{(9cos(4z)e^{(7)}ln(\frac{7}{4})\cdot(e^{(3)} - 1))}{146})dz=\frac{(9sin(4z)e^{(7)}ln(\frac{7}{4})\cdot(e^{(3)} - 1))}{584}|_{-6}^{-4}=-111.5\end{align*}$

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Example Question #332 : Triple Integrals

Evaluate: $\int_{0}^{1}\int_{0}^{2}\int_{0}^{3}x^3 \:dz \:dy\:dx$

Possible Answers:

$\frac{81}{2}$

$\frac{3}{2}$

$\frac{45}{4}$

$\frac{6}{5}$

Correct answer:

$\frac{3}{2}$

Explanation:

Evaluate $\int_{0}^{3}x^3 \:dz$ first.

$\int_{0}^{3}x^3 \:dz = x^3z|_{0}^{3} = 3x^3$

Substitute this back and evaluate $\int_{0}^{2}3x^3 \: dy$ .

$\int_{0}^{2}3x^3 \: dy = 3x^3y |_{0}^{2} = 6x^3$

Substitute this value and evaluate the last integral $\int_{0}^{1} 6x^3 dx$ .

$\int_{0}^{1} 6x^3 dx = \frac{6x^4}{4}|_{0}^{1}= \frac{3x^4}{2}|_{0}^{1} = \frac{3(1)^4}{2} = \frac{3}{2}$

The answer is: $\frac{3}{2}$

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Example Question #333 : Triple Integrals

Evaluate the following integral over the volume V which is enclosed by a cube of side length 2 in the first octant, with one corner on the origin.

$\int\int\int_V\frac{xyz}{8}\:dx\:dy\:dz$

Possible Answers:

$\frac{1}{8}$

$\frac{1}{64}$

Correct answer:

Explanation:

The description of the cube requires: 0<x<2, 0<y<2, and 0<z<2. The integral becomes:

$\int^2_0\int^2_0\int_0^2\frac{xyz}{8}\:dx\:dy\:dz$

To solve this integral:

$\int^2_0\int^2_0(\frac{x^2}{2}|^2_0)\frac{yz}{8}\:dy\:dz=\int^2_0\int^2_0\frac{yz}{4}\:dy\:dz$

$= \int^2_0(\frac{y^2}{2}|^2_0)\frac{z}{4}\:dz=\int^2_0\frac{z}{2}dz= \frac{z^2}{4}|^2_0=1$

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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 #324 : Multiple Integration

Example Question #325 : Multiple Integration

Example Question #326 : Multiple Integration

Example Question #327 : Multiple Integration

Example Question #328 : Multiple Integration

Example Question #329 : Multiple Integration

Example Question #330 : Multiple Integration

Example Question #331 : Triple Integrals

Example Question #332 : Triple Integrals

Example Question #333 : Triple Integrals

All Calculus 3 Resources

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