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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Calculate the value of the multiple integral.

34. H z 3 x 2 + y 2 + z 2 d V , where H is the solid hemisphere that lies above the xy-plane and has center the origin and radius 1

To determine

To calculate: The given triple integral.

Explanation

Given:

The function is f(x,y,z)=z3x2+y2+z2 .

The region H is the solid hemisphere centered at origin with radius 1 and lies above the xy-plane.

Formula used:

If f is a spherical region E given by aρb,αθβ,cϕd , then, Ef(x,y,z)dV=αβabcdf(ρsinϕcosθ,ρsinϕsinθ,ρcosϕ)ρ2sinϕdϕdρdθ (1)

If g(x) is the function of x and h(y) is the function of y and k(z) is the function of z  then, abcdefg(x)h(y)k(z)dzdydx=abg(x)dxcdh(y)dyefk(z)dz (2)

The spherical coordinates (ρ,θ,ϕ) corresponding to the rectangular coordinates (x,y,z) is,

ρ=x2+y2+z2ϕ=cos1(zρ)θ=cos1(xρsinϕ)

Calculation:

Since the equation of the sphere is given, solving with spherical coordinates is a wise choice. From the given conditions, ρ varies from 0 to 1, θ varies from 0 to 2π and ϕ varies from 0 to π2

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Chapter 15 Solutions

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