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Calculus 3 : Triple Integrals

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Example Questions

Example Question #348 : Triple Integration In Cartesian Coordinates

Evaluate the following integral

$\int \int \int_D 15xyz \ dV$ , where $D=[1,2]\times[3,4]\times [5,6]$

Possible Answers:

$\frac{3465}{8}$

$\frac{1}{8}$

$\frac{65}{8}$

Correct answer:

$\frac{3465}{8}$

Explanation:

$\int_{1}^{2} \int_{3}^{4} \int_{5}^{6} 15xyz \ dx\ dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} \frac{15}{2}x^2yz\Big|_{5}^{6} dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} (\frac{15}{2}(6)^2yz)-(\frac{15}{2}(5)^2yz) dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} \frac{165}{2}yz\ dy\ dz$

$=\int_{1}^{2} \frac{165}{4}y^2z\Big|_{3}^{4}\ dz$

$=\int_{1}^{2} \frac{165}{4}(4)^2z-\frac{165}{4}(3)^2z\ dz$

$=\int_{1}^{2} \frac{1155}{4}z\ dz$

$=\frac{1155}{8}z^2\Big|_{1}^{2}$

$=\frac{1155}{8}(2)^2-\frac{1155}{8}(1)^2=\frac{3465}{8}$

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Example Question #481 : Triple Integrals

Evaluate the following integral

$\int \int \int_D 15xyz \ dV$ , where $D=[1,2]\times[3,4]\times [5,6]$

Possible Answers:

$\frac{3465}{8}$

$\frac{65}{8}$

$\frac{1}{8}$

Correct answer:

$\frac{3465}{8}$

Explanation:

$\int_{1}^{2} \int_{3}^{4} \int_{5}^{6} 15xyz \ dx\ dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} \frac{15}{2}x^2yz\Big|_{5}^{6} dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} (\frac{15}{2}(6)^2yz)-(\frac{15}{2}(5)^2yz) dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} \frac{165}{2}yz\ dy\ dz$

$=\int_{1}^{2} \frac{165}{4}y^2z\Big|_{3}^{4}\ dz$

$=\int_{1}^{2} \frac{165}{4}(4)^2z-\frac{165}{4}(3)^2z\ dz$

$=\int_{1}^{2} \frac{1155}{4}z\ dz$

$=\frac{1155}{8}z^2\Big|_{1}^{2}$

$=\frac{1155}{8}(2)^2-\frac{1155}{8}(1)^2=\frac{3465}{8}$

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Example Question #1 : Triple Integration In Spherical Coordinates

Evaluate $\int \int \int_F 20z \ dV$ , where is the upper half of the sphere x^2+y^2+z^2=1 .

Possible Answers:

$\pi$

$2\pi$

$4\pi$

$5\pi$

Correct answer:

$5\pi$

Explanation:

Since we are only dealing with the upper half of a sphere, we can determine the boundaries easily, and remember to convert to spherical coordinates.

$0 \leq \rho \leq 1$

$0 \leq \theta \leq2\pi$

$0 \leq\phi\leq \frac{\pi}{2}$

$\int \int \int_F 20z \ dV=\int_{0}^{\frac{\pi}{2}} \int_{0}^{2\pi} \int_{0}^{1} \rho^2\sin(\phi)(20\rho\cos(\phi)) d\rho \ d\theta\ d\phi$

$=\int_{0}^{\frac{\pi}{2}} \int_{0}^{2\pi} \int_{0}^{1} 20\rho^3\sin(\phi)\cos(\phi) d\rho \ d\theta\ d\phi$

$=\int_{0}^{\frac{\pi}{2}} \int_{0}^{2\pi} 5\rho^4\sin(\phi)\cos(\phi) \Big|_{0}^{1} \ d\theta\ d\phi$

$=\int_{0}^{\frac{\pi}{2}} \int_{0}^{2\pi} 5(1)^4\sin(\phi)\cos(\phi)-0 \ d\theta\ d\phi$

$=\int_{0}^{\frac{\pi}{2}} \int_{0}^{2\pi} 5\sin(\phi)\cos(\phi) \ d\theta\ d\phi$

$=\int_{0}^{\frac{\pi}{2}} 5\theta \sin(\phi)\cos(\phi) \Big|_{0}^{2\pi} \ d\phi$

$=\int_{0}^{\frac{\pi}{2}} 10\pi \sin(\phi)\cos(\phi)-0 \ d\phi$

$=\int_{0}^{\frac{\pi}{2}} 10\pi \sin(\phi)\cos(\phi) \ d\phi$

$= -5\pi \cos^2(\phi)\Big|_{0}^{\frac{\pi}{2}}$

$= -5\pi \cos^2(\frac{\pi}{2})+5\pi \cos^2(0)=5\pi$

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Calculus 3 : Triple Integrals

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #348 : Triple Integration In Cartesian Coordinates

Example Question #481 : Triple Integrals

Example Question #1 : Triple Integration In Spherical Coordinates

All Calculus 3 Resources

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