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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Use cylindrical or spherical coordinates, whichever seems more appropriate.

36. Find the volume of the smaller wedge cut from a sphere of radius a by two planes that intersect along a diameter at an angle of π/6.

To determine

To find: The volume of the given region by using spherical or cylindrical coordinates.

Explanation

Given:

The region B is a small wedge cut from a sphere of radius a by two planes that intersect along a diameter at an angle of π6 .

Formula used:

The volume of the given region B is, BdV .

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, it is more appropriate to use the spherical coordinates. By the given conditions, it is observed that ρ varies from 0 to a, θ varies from 0 to π6 and ϕ varies from 0 to π . Use the formula mentioned above to find the volume of the given region. Then, by equation (1), the volume of the given region is,

BdV=0π60π0adzdydx=0π60π0a(ρ2sinϕ)dρdϕdθ

Integrate with respect to ρ and apply the limit of it

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