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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #221 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{9}^{11}\int_{4}^{7}\int_{-9}^{-7}(\frac{(2cos(z + 2)e^{(2y)})}{(13x^{2})})dxdydz\end{align*}$

Possible Answers:

4160

16644

Correct answer:

4160

Explanation:

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

$\begin{align*}&\int_{9}^{11}\int_{4}^{7}\int_{-9}^{-7}(\frac{(2cos(z + 2)e^{(2y)})}{(13x^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{9}^{11}\int_{4}^{7}\int_{-9}^{-7}(\frac{(2cos(z + 2)e^{(2y)})}{(13x^{2})})dxdydz=\int_{9}^{11}\int_{4}^{7}(-\frac{(2cos(z + 2)e^{(2y)})}{(13x)})dydz|_{-9}^{-7}\\&\int_{9}^{11}\int_{4}^{7}(\frac{(4cos(z + 2)e^{(2y)})}{819})dydz=\int_{9}^{11}(\frac{(2cos(z + 2)e^{(2y)})}{819})dz|_{4}^{7}\\&\int_{9}^{11}(\frac{(2cos(z + 2)e^{(8)}\cdot (e^{(6)} - 1))}{819})dz=\frac{(2sin(z + 2)e^{(8)}\cdot (e^{(6)} - 1))}{819}|_{9}^{11}=4160\end{align*}$

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Example Question #221 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{11}^{13}\int_{-3.5}^{-1.5}\int_{-4}^{1}(\frac{(11\cdot 2^{(\frac{x}{4})}e^{(-2y)})}{(8z)})dxdydz\end{align*}$

Possible Answers:

-98.35

122.9

491.8

Correct answer:

491.8

Explanation:

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

$\begin{align*}&\int_{11}^{13}\int_{-3.5}^{-1.5}\int_{-4}^{1}(\frac{(11\cdot 2^{(\frac{x}{4})}e^{(-2y)})}{(8z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{11}^{13}\int_{-3.5}^{-1.5}\int_{-4}^{1}(\frac{(11\cdot 2^{(\frac{x}{4})}e^{(-2y)})}{(8z)})dxdydz=\int_{11}^{13}\int_{-3.5}^{-1.5}(\frac{(11\cdot 2^{(\frac{x}{4})}e^{(-2y)})}{(2zln(2))})dydz|_{-4}^{1}\\&\int_{11}^{13}\int_{-3.5}^{-1.5}(\frac{(11e^{(-2y)}\cdot (2\cdot 2^{(\frac{1}{4})} - 1))}{(4zln(2))})dydz=\int_{11}^{13}(-\frac{(11e^{(-2y)}\cdot (2\cdot 2^{(\frac{1}{4})} - 1))}{(8zln(2))})dz|_{-3.5}^{-1.5}\\&\int_{11}^{13}(\frac{(11e^{(3)} - 11e^{(7)} - 22\cdot 2^{(\frac{1}{4})}e^{(3)} + 22\cdot 2^{(\frac{1}{4})}e^{(7)})}{(8zln(2))})dz=\frac{(ln(z)\cdot (11e^{(3)} - 11e^{(7)} - 22\cdot 2^{(\frac{1}{4})}e^{(3)} + 22\cdot 2^{(\frac{1}{4})}e^{(7)}))}{(8ln(2))}|_{11}^{13}=491.8\end{align*}$

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Example Question #351 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-7.5}\int_{-4.5}^{-2.5}\int_{4}^{5}(\frac{(3^{(\frac{x}{4})}cos(z + 2))}{(26\cdot 3^y)})dxdydz\end{align*}$

Possible Answers:

102.7

-4.11

-123.2

20.54

Correct answer:

20.54

Explanation:

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

$\begin{align*}&\int_{-9}^{-7.5}\int_{-4.5}^{-2.5}\int_{4}^{5}(\frac{(3^{(\frac{x}{4})}cos(z + 2))}{(26\cdot 3^y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-7.5}\int_{-4.5}^{-2.5}\int_{4}^{5}(\frac{(3^{(\frac{x}{4})}cos(z + 2))}{(26\cdot 3^y)})dxdydz=\int_{-9}^{-7.5}\int_{-4.5}^{-2.5}(\frac{(2\cdot (\frac{1}{3})^y\cdot 3^{(\frac{x}{4})}cos(z + 2))}{(13ln(3))})dydz|_{4}^{5}\\&\int_{-9}^{-7.5}\int_{-4.5}^{-2.5}(\frac{(6cos(z + 2)\cdot (3^{(\frac{1}{4})} - 1))}{(13\cdot 3^yln(3))})dydz=\int_{-9}^{-7.5}(\frac{(6\cdot (cos(z + 2) - 3^{(\frac{1}{4})}cos(z + 2)))}{(13\cdot 3^yln(3)^{2})})dz|_{-4.5}^{-2.5}\\&\int_{-9}^{-7.5}(-\frac{(432\cdot 3^{(\frac{1}{2})}cos(z + 2) - 432\cdot 3^{(\frac{3}{4})}cos(z + 2))}{(13ln(3)^{2})})dz=-\frac{(432\cdot 3^{(\frac{1}{2})}sin(z + 2) - 432\cdot 27^{(\frac{1}{4})}sin(z + 2))}{(13ln(3)^{2})}|_{-9}^{-7.5}=20.54\end{align*}$

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Example Question #222 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-4}^{-1.5}\int_{-4}^{0.5}\int_{-9}^{-4.5}(\frac{(43x^{2}e^{(y)}e^{(z)})}{12})dxdydz\end{align*}$

Possible Answers:

-42.4

1272

254.4

Correct answer:

254.4

Explanation:

$\begin{align*}&\int_{-4}^{-1.5}\int_{-4}^{0.5}\int_{-9}^{-4.5}(\frac{(43x^{2}e^{(y)}e^{(z)})}{12})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4}^{-1.5}\int_{-4}^{0.5}\int_{-9}^{-4.5}(\frac{(43x^{2}e^{(y)}e^{(z)})}{12})dxdydz=\int_{-4}^{-1.5}\int_{-4}^{0.5}(\frac{(43x^{3}e^{(y + z)})}{36})dydz|_{-9}^{-4.5}\\&\int_{-4}^{-1.5}\int_{-4}^{0.5}(\frac{(24381e^{(y + z)})}{32})dydz=\int_{-4}^{-1.5}(\frac{(24381e^{(y + z)})}{32})dz|_{-4}^{0.5}\\&\int_{-4}^{-1.5}(\frac{(24381e^{(-4)}e^{(z)}\cdot (e^{(\frac{9}{2})} - 1))}{32})dz=e^{(z - 4)}\cdot (\frac{(24381e^{(\frac{9}{2})})}{32}-\frac{ 24381}{32})|_{-4}^{-1.5}=254.4\end{align*}$

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Example Question #223 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{5.5}\int_{4}^{5}\int_{-6}^{-1.5}(\frac{(3\cdot 3^{(\frac{z}{2})}\cdot 3^y)}{(74x^{2})})dxdydz\end{align*}$

Possible Answers:

83.36

250.1

-333.4

-16.67

Correct answer:

83.36

Explanation:

$\begin{align*}&\int_{3}^{5.5}\int_{4}^{5}\int_{-6}^{-1.5}(\frac{(3\cdot 3^{(\frac{z}{2})}\cdot 3^y)}{(74x^{2})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{5.5}\int_{4}^{5}\int_{-6}^{-1.5}(\frac{(3\cdot 3^{(\frac{z}{2})}\cdot 3^y)}{(74x^{2})})dxdydz=\int_{3}^{5.5}\int_{4}^{5}(-\frac{(3\cdot 3^{(y +\frac{ z}{2})})}{(74x)})dydz|_{-6}^{-1.5}\\&\int_{3}^{5.5}\int_{4}^{5}(\frac{(3\cdot 3^{(y +\frac{ z}{2})})}{148})dydz=\int_{3}^{5.5}(\frac{(3\cdot 3^{(y +\frac{ z}{2})})}{(148ln(3))})dz|_{4}^{5}\\&\int_{3}^{5.5}(\frac{(243\cdot (3^{(\frac{1}{2})})^z)}{(74ln(3))})dz=\frac{(243\cdot 3^{(\frac{z}{2})})}{(37ln(3)^{2})}|_{3}^{5.5}=83.36\end{align*}$

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Example Question #351 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-7}\int_{-6}^{-3.5}\int_{6}^{11}(\frac{cos(x + 1)}{78})dxdydz\end{align*}$

Possible Answers:

0.08

-0.31

-0.08

0.02

Correct answer:

-0.08

Explanation:

$\begin{align*}&\int_{-9}^{-7}\int_{-6}^{-3.5}\int_{6}^{11}(\frac{cos(x + 1)}{78})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-7}\int_{-6}^{-3.5}\int_{6}^{11}(\frac{cos(x + 1)}{78})dxdydz=\int_{-9}^{-7}\int_{-6}^{-3.5}(\frac{sin(x + 1)}{78})dydz|_{6}^{11}\\&\int_{-9}^{-7}\int_{-6}^{-3.5}(\frac{sin(12)}{78}-\frac{ sin(7)}{78})dydz=\int_{-9}^{-7}(-y\cdot (\frac{sin(7)}{78}-\frac{ sin(12)}{78}))dz|_{-6}^{-3.5}\\&\int_{-9}^{-7}(\frac{(5sin(12))}{156}-\frac{ (5sin(7))}{156})dz=-z\cdot (\frac{(5sin(7))}{156}-\frac{ (5sin(12))}{156})|_{-9}^{-7}=-0.08\end{align*}$

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Example Question #360 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{4.5}^{6.5}\int_{-5}^{-1.5}\int_{4}^{7.5}(\frac{(41\cdot 3^z)}{(4\cdot 3^{(\frac{x}{2})}\cdot 3^{(\frac{y}{2})})})dxdydz\end{align*}$

Possible Answers:

14611

43822

Correct answer:

43822

Explanation:

$\begin{align*}&\int_{4.5}^{6.5}\int_{-5}^{-1.5}\int_{4}^{7.5}(\frac{(41\cdot 3^z)}{(4\cdot 3^{(\frac{x}{2})}\cdot 3^{(\frac{y}{2})})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4.5}^{6.5}\int_{-5}^{-1.5}\int_{4}^{7.5}(\frac{(41\cdot 3^z)}{(4\cdot 3^{(\frac{x}{2})}\cdot 3^{(\frac{y}{2})})})dxdydz=\int_{4.5}^{6.5}\int_{-5}^{-1.5}(-\frac{(41\cdot 3^z)}{(2\cdot 3^{(\frac{x}{2})}\cdot 3^{(\frac{y}{2})}ln(3))})dydz|_{4}^{7.5}\\&\int_{4.5}^{6.5}\int_{-5}^{-1.5}(-\frac{(3^{(z -\frac{ y}{2})}\cdot (\frac{(41\cdot 3^{(\frac{1}{4})})}{162}-\frac{ 41}{18}))}{ln(3)})dydz=\int_{4.5}^{6.5}(\frac{(41\cdot 3^{(z -\frac{ y}{2})}\cdot (3^{(\frac{1}{4})} - 9))}{(81ln(3)^{2})})dz|_{-5}^{-1.5}\\&\int_{4.5}^{6.5}(\frac{(41\cdot 3^z\cdot (27\cdot 3^{(\frac{1}{2})} - 6\cdot 3^{(\frac{3}{4})} + 1))}{(27ln(3)^{2})})dz=\frac{(41\cdot 3^z\cdot (27\cdot 3^{(\frac{1}{2})} - 6\cdot 3^{(\frac{3}{4})} + 1))}{(27ln(3)^{3})}|_{4.5}^{6.5}=43822\end{align*}$

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Example Question #361 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{3.5}^{8.5}\int_{-5}^{-1.5}\int_{-4}^{2}(\frac{(58e^{(x)})}{(7\cdot 2^y\cdot 3^{(\frac{z}{2})})})dxdydz\end{align*}$

Possible Answers:

-640.3

2561

640.3

-128.1

Correct answer:

640.3

Explanation:

$\begin{align*}&\int_{3.5}^{8.5}\int_{-5}^{-1.5}\int_{-4}^{2}(\frac{(58e^{(x)})}{(7\cdot 2^y\cdot 3^{(\frac{z}{2})})})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3.5}^{8.5}\int_{-5}^{-1.5}\int_{-4}^{2}(\frac{(58e^{(x)})}{(7\cdot 2^y\cdot 3^{(\frac{z}{2})})})dxdydz=\int_{3.5}^{8.5}\int_{-5}^{-1.5}(\frac{(58e^{(x)})}{(7\cdot 2^y\cdot 3^{(\frac{z}{2})})})dydz|_{-4}^{2}\\&\int_{3.5}^{8.5}\int_{-5}^{-1.5}(\frac{(58\cdot (e^{(2)} - e^{(-4)}))}{(7\cdot 2^y\cdot 3^{(\frac{z}{2})})})dydz=\int_{3.5}^{8.5}(-\frac{(58e^{(-4)}\cdot (e^{(6)} - 1))}{(7\cdot 2^y\cdot 3^{(\frac{z}{2})}ln(2))})dz|_{-5}^{-1.5}\\&\int_{3.5}^{8.5}(\frac{(1856e^{(2)} - 1856e^{(-4)} - 116\cdot 2^{(\frac{1}{2})}e^{(2)} + 116\cdot 2^{(\frac{1}{2})}e^{(-4)})}{(7\cdot 3^{(\frac{z}{2})}ln(2))})dz=-\frac{(232e^{(-4)}\cdot (16e^{(6)} - 2^{(\frac{1}{2})}e^{(6)} + 2^{(\frac{1}{2})} - 16))}{(7\cdot 3^{(\frac{z}{2})}ln(2)ln(3))}|_{3.5}^{8.5}=640.3\end{align*}$

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Example Question #832 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{5}^{10}\int_{-6}^{-4}\int_{7}^{8.5}(\frac{(2\cdot 3^z)}{(7xy)})dxdydz\end{align*}$

Possible Answers:

602

4816

Correct answer:

Explanation:

$\begin{align*}&\int_{5}^{10}\int_{-6}^{-4}\int_{7}^{8.5}(\frac{(2\cdot 3^z)}{(7xy)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{5}^{10}\int_{-6}^{-4}\int_{7}^{8.5}(\frac{(2\cdot 3^z)}{(7xy)})dxdydz=\int_{5}^{10}\int_{-6}^{-4}(\frac{(2\cdot 3^zln(x))}{(7y)})dydz|_{7}^{8.5}\\&\int_{5}^{10}\int_{-6}^{-4}(\frac{(2\cdot 3^zln(\frac{17}{14}))}{(7y)})dydz=\int_{5}^{10}(\frac{(2\cdot 3^zln(\frac{17}{14})ln(y))}{7})dz|_{-6}^{-4}\\&\int_{5}^{10}(\frac{(2\cdot 3^zln(\frac{2}{3})ln(\frac{17}{14}))}{7})dz=\frac{(2\cdot 3^zln(\frac{2}{3})ln(\frac{17}{14}))}{(7ln(3))}|_{5}^{10}=-1204\end{align*}$

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Example Question #221 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-4.5}^{-3.5}\int_{6}^{10}\int_{10}^{15}(\frac{(26cos(y + 2))}{(9\cdot 3^z)})dxdydz\end{align*}$

Possible Answers:

9382

469.1

Correct answer:

Explanation:

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

$\begin{align*}&\int_{-4.5}^{-3.5}\int_{6}^{10}\int_{10}^{15}(\frac{(26cos(y + 2))}{(9\cdot 3^z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4.5}^{-3.5}\int_{6}^{10}\int_{10}^{15}(\frac{(26cos(y + 2))}{(9\cdot 3^z)})dxdydz=\int_{-4.5}^{-3.5}\int_{6}^{10}(\frac{(26xcos(y + 2))}{(9\cdot 3^z)})dydz|_{10}^{15}\\&\int_{-4.5}^{-3.5}\int_{6}^{10}(\frac{(130cos(y + 2))}{(9\cdot 3^z)})dydz=\int_{-4.5}^{-3.5}(\frac{(130sin(y + 2))}{(9\cdot 3^z)})dz|_{6}^{10}\\&\int_{-4.5}^{-3.5}(-\frac{(130\cdot (sin(8) - sin(12)))}{(9\cdot 3^z)})dz=\frac{(130\cdot (sin(8) - sin(12)))}{(9\cdot 3^zln(3))}|_{-4.5}^{-3.5}=-1876\end{align*}$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #221 : Triple Integration In Cartesian Coordinates

Example Question #221 : Triple Integration In Cartesian Coordinates

Example Question #351 : Multiple Integration

Example Question #222 : Triple Integration In Cartesian Coordinates

Example Question #223 : Triple Integration In Cartesian Coordinates

Example Question #351 : Triple Integrals

Example Question #360 : Multiple Integration

Example Question #361 : Multiple Integration

Example Question #832 : Calculus 3

Example Question #221 : Triple Integration In Cartesian Coordinates

All Calculus 3 Resources

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