Evaluate the integrals (x + y + z) dy dx dzJo Jo

To evaluate the integral: ∫-11∫01∫02x+y+zdx dy dz Evaluating the integral: ∫-11∫01∫02x+y+zdx dy…

Answered: (x + y + z) dy dx dz Jo Jo

Elementary Algebra

17th Edition

ISBN:9780998625713

Author:Lynn Marecek, MaryAnne Anthony-Smith

Publisher:Lynn Marecek, MaryAnne Anthony-Smith

Chapter8: Rational Expressions And Equations

Section8.9: Use Direct And Inverse Variation

Problem 498E: If y varies inversely with x and y=11 when x=3 find the equation that relates x and y.

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Evaluate the integrals

Expert Solution

Step 1

To evaluate the integral: $\int_{- 1}^{1} \int_{0}^{1} \int_{0}^{2} (x + y + z) d x d y d z$

Evaluating the integral:

$\begin{array}{rcl} \int_{- 1}^{1} \int_{0}^{1} \int_{0}^{2} (x + y + z) d x d y d z & = & \int_{- 1}^{1} \int_{0}^{1} [\int_{0}^{2} (x + y + z) d x] d y d z \\ = & \int_{- 1}^{1} \int_{0}^{1} {[\frac{x^{2}}{2} + x y + x z]}_{0}^{2} d y d z \\ = & \int_{- 1}^{1} \int_{0}^{1} [2 + 2 y + 2 z - 0] d y d z \\ = & 2 \int_{- 1}^{1} \int_{0}^{1} (1 + y + z) d y d z \\ = & 2 \int_{- 1}^{1} {(y + \frac{y^{2}}{2} + y z)}_{0}^{1} d z \\ = & 2 \int_{- 1}^{1} (1 + \frac{1}{2} + z - 0) d z \\ = & 2 \int_{- 1}^{1} (\frac{3}{2} + z) d z \\ = & 2 {(\frac{3}{2} z + \frac{z^{2}}{2})}_{- 1}^{1} \\ = & (3 + 1 - 3 - 1) \\ = & 0 \end{array}$

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