3. Use spherical coordinates to valuate fff, y? dV where E is enclosed by the hemisphere x? + y? + z? < 9 and y >

Given that ∫∫∫Ey2dV and enclosed by the hemisphere, x2+y2+z2≤9,y≥0 Here to evaluate the integral,…

Answered: 3. Use spherical coordinates to valuate…

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter1: Line And Angle Relationships

Section1.2: Angles And Their Relationships

Problem 9E: Use drawings as needed to answer each of the following questions. Must two rays with a common...

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3. Use spherical coordinates to valuate fff, y? dV where E is enclosed by the hemisphere x? + y? + z? < 9 and y >

Expert Solution

Step 1

Given that

$\int \int \int_{E} y^{2} d V$ and enclosed by the hemisphere, $x^{2} + y^{2} + z^{2} \leq 9, y \geq 0$

Here to evaluate the integral, convert the cartesian coordinates into Spherical coordinates as

$x = ρ \cos θ \cos ϕ y = ρ \sin θ \sin ϕ z = ρ \cos ϕ$

To find $ρ$ :

$x^{2} + y^{2} + z^{2} = ρ^{2} : d V = ρ^{2} \sin ϕ d ρ d θ d ϕ x^{2} + y^{2} + z^{2} = 9 = 3^{2} ρ = 3$

The volume of the region between the spheres is given by,

$V = \int_{c}^{d} \int_{α}^{β} \int_{a}^{b} f (ρ \sin ϕ \cos θ, ρ \sin ϕ \sin θ, ρ \cos ϕ) ρ^{2} \sin ϕ d ρ d θ d ϕ$

Where E is the spherical wedge given by

$E = \{(ρ, θ, ϕ) | a \leq ρ \leq b, α \leq θ \leq β, c \leq ϕ \leq d\}$

The region is defined by

$E = \{(ρ, θ, ϕ) | 0 \leq ρ \leq 3, \frac{- π}{2} \leq θ \leq \frac{π}{2}, 0 \leq ϕ \leq π\}$

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3. Use spherical coordinates to valuate ff, y? dV where E is enclosed by the hemisphere x? + y? + z? < 9 and y 2