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數學難題 integrals

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Find the volume of the solid: (i) Below the graph of f( x, y) = 25 - x^2 - y^2 and above the plane z =16 (ii) Bounded by the upper hemisphere x^2+y^2+z^2=8, z greater and equal 0 and the cone z=square root(x^2+y^2)

最佳解答:

(i) At any z, the graph f(x,y)=25 - x^2 - y^2 is a circle of radius r, where r^2 = 25 - z. The volume of the disc bounded by the plan z and z + dz is PI(r^2)dz Therefore volume = Integral z=16 to z=25 [PI * (25 - z)]dz = [PI * (25z - z^2/2)] z=16 to z=25 = PI * (25 * 25 - 25 * 25/2 - 25 * 16 + 16 * 16/2) = 40.5 * PI = 127.23 (ii) we need to find the intersection of the sphere and the cone as given by: x^2 + y^2 + z^2 = 8 ...(1) z^2 = x^2 + y^2 ...(2) (1)&(2) yields z = 2 in upper hemisphere When z = 2, x^2 + y^2 = 4 Volume required = volume of the cone + the part of the sphere above z=2 Volume of the cone = 1/3 * PI * r^2 * h = 1/3 * PI * 4 * 2 = 8/3 * PI = 8.38 Volume of the part of the sphere above z = 2 is Integral z=2 to z=sqrt(8) PI * r^2 dz And r^2 = 8 - z^2 The integral becomes Integral z=2 to z=sqrt(8) [PI * (8 - z^2)]dz = PI * (8z - z^3/3) z=2 to z=sqrt(8) = PI * (8 * sqrt(8) - sqrt(8)^3/3 - 8 * 2 + 2^3/3) = 5.50 Total volume = 13.88

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