Chapter 13: Q. 27 (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.
$\int_{0}^{4} \int_{0}^{3 - \frac{3}{4} x} \int_{0}^{2 - \frac{1}{2} x - \frac{2}{3} y} d z d y d x$

Short Answer

Expert verified

$\int_{0}^{4} \int_{0}^{3 - \frac{3}{4} x} \int_{0}^{2 - \frac{1}{2} x - \frac{2}{3} y} d z d y d x = 4$

Step by step solution

Given

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

$\int_{0}^{4} \int_{0}^{3 - \frac{3}{4} x} \int_{0}^{2 - \frac{1}{2} x - \frac{2}{3} y} d z d y d x$

Evaluate the integral

$V = \int_{0}^{4} \int_{0}^{3 - \frac{3}{4} x} [\int_{0}^{2 - \frac{1}{2} x - \frac{2}{3} y} d z] d y d x V = \int_{0}^{4} [\int_{0}^{3 - \frac{3}{4} x} (2 - \frac{1}{2} x - \frac{2}{3} y) d y] d x V = \int_{0}^{4} {[2 y - \frac{1}{2} x y - \frac{2}{3} \frac{y^{2}}{2}]}_{0}^{3 - \frac{3}{4} x} V = \int_{0}^{4} [2 (3 - \frac{3}{4} x) - \frac{1}{2} x (3 - \frac{3}{4} x) - \frac{1}{3} {(3 - \frac{3}{4} x)}^{2}] V = \int_{0}^{4} (3 - \frac{3}{2} x + \frac{3}{16} x^{2}) d x V = {[3 x - \frac{3}{2} \frac{x^{2}}{2} + \frac{3}{16} \frac{x^{3}}{3}]}_{0}^{4} V = [12 - 12 + 4] V = 4 cubic units$

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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.
$\int_{0}^{4} \int_{0}^{3 - \frac{3}{4} x} \int_{0}^{2 - \frac{1}{2} x - \frac{2}{3} y} d z d y d x$

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.∫04∫03-34x∫02-12x-23ydzdydx

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

Logic and Functions

Mechanics Maths

Theoretical and Mathematical Physics

Pure Maths

Statistics

Probability and Statistics

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.
$\int_{0}^{4} \int_{0}^{3 - \frac{3}{4} x} \int_{0}^{2 - \frac{1}{2} x - \frac{2}{3} y} d z d y d x$