Change the integral into cylindrical coordinates.4-a28-22-y2(22²+y²) dzdyda

Answered: 4-a2 8-a2-y2 (2² + y²) dzdydx 2

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Change the integral into cylindrical coordinates.
4-a2
8-22-y2
(22
²+y²) dzdyda

Expert Solution

Step 1

Given

$\int_{- 2}^{2} \int_{0}^{\sqrt{4 - x^{2}}} \int_{\sqrt{x^{2} + y^{2}}}^{\sqrt{8 - x^{2} - y^{2}}} (x^{2} + y^{2}) d z d y d x$

Change the integral into cylindrical coordinates

Step 2

let

$x = 2 \cos θ y = 2 \sin θ where 0 < θ \leq π z = z$

Then

$0 < r < 2$

since $0 < y < \sqrt{4 - x^{2}}$ this implies it is a semicircle in a half xy-plane with y>0

therefore $0 < θ \leq π$

$\sqrt{x^{2} + y^{2}} = \sqrt{r^{2} \cos^{2} θ + r^{2} \sin^{2} θ} \sqrt{x^{2} + y^{2}} = r \sqrt{\cos^{2} θ + \sin^{2} θ} \sqrt{x^{2} + y^{2}} = r \sqrt{1} \sqrt{x^{2} + y^{2}} = r$

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