Problem. Use the spherical coordinatas, I= psin o cos 0, y= psin osin 0, z= pcos o; p = r +y° +z, dV = p° sin o dp dộ do, to calculate the intogral I. "av, where E is the solid hemisphere r? + y? + z <1, z 2 0. SII,-av. dV,

Problem. Use the spherical coordinatas, I= psin o cos 0, y= psin osin 0, z= pcos o; p = r +y° +z, dV = p° sin o dp dộ do, to calculate the intogral I. "av, where E is the solid hemisphere r? + y? + z <1, z 2 0. SII,-av. dV,

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section: Chapter Questions

Problem 17T

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Problem. Use the spherical coordinatas,
I= psin o cos 0, y= psin osin 0, z= pcos o;
p = r +y° +z, dV = p° sin o dp dộ do,
to calculate the intogral
I. "av,
where E is the solid hemisphere r? + y? + z <1, z 2 0.
SII,-av.
dV,

Expert Solution

Step 1

Let us first analyse E to get the limit values of integration.

We are given that E is the solid hemisphere x2+y2+z2 ≤ 1 and z ≥ 0.

This hemisphere is centered at the origin and is considered only in the positive direction of z.

We have taken,

So, by the above conditions, the region of integration E in spherical coordinates is described by the inequalities

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