Chapter 13: Q. 28 (page 1082)

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
$\int_{0}^{3} \int_{0}^{2 - \frac{2}{3} y} \int_{0}^{4 - \frac{4}{3} y - 2 z} d x d z d y$

Short Answer

Expert verified

$\int_{0}^{3} \int_{0}^{2 - \frac{2}{3} y} \int_{0}^{4 - \frac{4}{3} y - 2 z} d x d z d y = 4$

Step by step solution

Given

We have to find the volume by using the triple integral.

\int_{0}^{3} \int_{0}^{2 - \frac{2}{3} y} \int_{0}^{4 - \frac{4}{3} y - 2 z} d x d z d y

Evaluate the integral

$V = \int_{0}^{3} \int_{0}^{2 - \frac{2}{3} y} \int_{0}^{4 - \frac{4}{3} y - 2 z} d x d z d y V = \int_{0}^{3} [\int_{0}^{2 - \frac{2}{3} y} (4 - \frac{4}{3} y - 2 z) d z] d y V = \int_{0}^{3} {[4 z - \frac{4}{3} y z - z^{2}]}_{0}^{2 - \frac{2}{3} y} d y V = \int_{0}^{3} [4 (2 - \frac{2}{3} y) - \frac{4}{3} y (2 - \frac{2}{3} y) - {(2 - \frac{2}{3} y)}^{2}] d y V = \int_{0}^{3} (4 - \frac{8}{3} y + \frac{4}{9} y^{2}) d y V = {[4 y - \frac{8}{3} \frac{y^{2}}{2} + \frac{4}{9} \frac{y^{3}}{3}]}_{0}^{3} V = [12 - 12 + 4] V = 4 cubic units$

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Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
$\int_{0}^{3} \int_{0}^{2 - \frac{2}{3} y} \int_{0}^{4 - \frac{4}{3} y - 2 z} d x d z d y$

Short Answer

Step by step solution

Given

Evaluate the integral

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Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.∫03∫02-23y∫04-43y-2zdxdzdy

Short Answer

Step by step solution

Given

Evaluate the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Logic and Functions

Probability and Statistics

Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
$\int_{0}^{3} \int_{0}^{2 - \frac{2}{3} y} \int_{0}^{4 - \frac{4}{3} y - 2 z} d x d z d y$