Convert the integral\[\int_{0}^{3} \int_{0}^{\sqrt{9-x^2}} \int_{0}^{\sqrt{x^2+y^2}} 2xyz \, dz \, dy \, dx\]into an integral in cylindrical coordinates. You do not have to evaluate the integral after you convert it.

Answered: Convert the integral 2x42 dz dx into an…

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

Convert the integral

\[
\int_{0}^{3} \int_{0}^{\sqrt{9-x^2}} \int_{0}^{\sqrt{x^2+y^2}} 2xyz \, dz \, dy \, dx
\]

into an integral in cylindrical coordinates. You do not have to evaluate the integral after you convert it.

Expert Solution

Step 1

$w e c o n v e r t t h e r e c \tan g u l a r c o - o r d i n a t e s i n t o c y l i n d r i c a l c o - o r d i n a t e s \int \int \int f (x, y, z) d x d y d z = \int \int \int f (r \cos θ, r \sin θ) r d z d r d θ p u t x = r \cos θ, y = r \sin θ a n d z = z g i v e n \int_{0}^{3} \int_{0}^{\sqrt{9 - x^{2}}} \int_{0}^{\sqrt{x^{2} + y^{2}}} 2 x y z d z d y d x = \int_{0}^{π} \int_{0}^{3} \int_{0}^{r} 2 r \cos θ r \sin θ z r d z d r d θ = \int_{0}^{π} \int_{0}^{3} \int_{0}^{r} r^{3} \sin 2 θ z d z d r d θ$

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