Chapter 13: Q. 2 (page 1017)

Evaluate the triple integral $\int_{0}^{2} \int_{0}^{- \frac{3}{2} x + 3} \int_{0}^{4 - 2 x - \frac{4}{3} y} d z d y d x$ and given physical interpretation of the integral

Short Answer

Expert verified

The value of the triple integral is $4 c u b i c u n i t s .$ it represents the volume of the rectangular region.

Step by step solution

Given information

We are given a triple integral $\int_{0}^{2} \int_{0}^{- \frac{3}{2} x + 3} \int_{0}^{4 - 2 x - \frac{4}{3} y} d z d y d x$

Evaluate the integral

We get,

$\int_{0}^{2} \int_{0}^{- \frac{3}{2} x + 3} \int_{0}^{4 - 2 x - \frac{4}{3} y} d z d y d x = \int_{0}^{2} \int_{0}^{- \frac{3}{2} x + 3} [4 - 2 x - \frac{4}{3} y] d y d x = \int_{0}^{2} {[4 y - 2 x y - \frac{2}{3} y^{2}]}_{0}^{- \frac{3}{2} x + 3} d x = \int_{0}^{2} [- 6 x + 12 + 3 x^{2} - 6 x - \frac{3}{2} x^{2} - 6 + 6 x] d x = \int_{0}^{2} [\frac{3}{2} x^{2} - 6 x + 6] d x = {[\frac{1}{2} x^{3} - 3 x^{2} + 6 x]}_{0}^{2} = 4 = 4 c u b i c u n i t s$

It represents the volume of the rectangular region

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Evaluate the triple integral $\int_{0}^{2} \int_{0}^{- \frac{3}{2} x + 3} \int_{0}^{4 - 2 x - \frac{4}{3} y} d z d y d x$ and given physical interpretation of the integral

Short Answer

Step by step solution

Given information

Evaluate the integral

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Evaluate the triple integral∫02∫0-32x+3∫04-2x-43ydzdydxand given physical interpretation of the integral

Short Answer

Step by step solution

Given information

Evaluate the integral

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Most popular questions from this chapter

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Evaluate the triple integral $\int_{0}^{2} \int_{0}^{- \frac{3}{2} x + 3} \int_{0}^{4 - 2 x - \frac{4}{3} y} d z d y d x$ and given physical interpretation of the integral