I need your help in this problem Sure! Here’s the transcription:---Use spherical coordinates to compute \[ I = \iiint_W \frac{dv}{\sqrt{x^2 + y^2 + z^2}} , \]where $ W $ is the solid bounded by the spheres \[ x^2 + y^2 + z^2 = 9 \] and \[ x^2 + y^2 + z^2 = 4. \]\[ I = \, ? \]--- This text describes how to evaluate a triple integral using spherical coordinates. The solid $ W $ is defined by the region between two concentric spheres with radii 3 and 2. The goal is to compute the integral of $\frac{1}{\sqrt{x^2 + y^2 + z^2}}$ over the volume $ W $.

Answered: dv Use spherical Coordinares to compute…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

I need your help in this problem

Sure! Here’s the transcription:

---

Use spherical coordinates to compute

\[ I = \iiint_W \frac{dv}{\sqrt{x^2 + y^2 + z^2}} , \]

where $ W $ is the solid bounded by the spheres

\[ x^2 + y^2 + z^2 = 9 \]

and

\[ x^2 + y^2 + z^2 = 4. \]

\[ I = \, ? \]

---

This text describes how to evaluate a triple integral using spherical coordinates. The solid $ W $ is defined by the region between two concentric spheres with radii 3 and 2. The goal is to compute the integral of $\frac{1}{\sqrt{x^2 + y^2 + z^2}}$ over the volume $ W $.

Expert Solution

Step 1

Given

$I = \int \int \int_{w} \frac{d V}{\sqrt{x^{2} + y^{2} + z^{2}}} W h e r e W i s t h e s o l i d b o u n d e d b y t h e s p h e r e s x^{2} + y^{2} + z^{2} = 9 a n d x^{2} + y^{2} + z^{2} = 4 .$

Step 2

Now

$U s e t h e s p h e r i c a l c o o r d i n a t e s x = ρ \cos θ \sin ϕ y = ρ \sin θ \sin ϕ z = ρ \cos ϕ a n d x^{2} + y^{2} + z^{2} = {(ρ \cos θ \sin ϕ)}^{2} + {(ρ \sin θ \sin ϕ)}^{2} + {(ρ \cos ϕ)}^{2} x^{2} + y^{2} + z^{2} = ρ^{2} [{(\sin ϕ)}^{2} ({(\cos θ)}^{2} + {(\sin θ)}^{2}) + {(\cos ϕ)}^{2}] x^{2} + y^{2} + z^{2} = ρ^{2} [{(\sin ϕ)}^{2} (1) + {(\cos ϕ)}^{2}] x^{2} + y^{2} + z^{2} = ρ^{2} [1] \{i f w e k n o w \cos^{2} x + \sin^{2} x = 1\} t h e n w e h a v e x^{2} + y^{2} + z^{2} = ρ^{2} a n d g i v e n x^{2} + y^{2} + z^{2} = 9, x^{2} + y^{2} + z^{2} = 4 ρ^{2} = 9, ρ^{2} = 4 ρ = 3, ρ = 2 S o t h e l i m i t i s 2 \leq ρ \leq 3 0 \leq ϕ \leq π 0 \leq θ \leq 2 π$

Step by step

Solved in 3 steps

Knowledge Booster

$Research Ethics$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions