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標題:

ddfinite integration

發問:

Show thatthe volume of a sphere with the radius of r is (4/3) πr^3 by definiteintegration. Consider the equation (x^2)+(y^2)=r^2

最佳解答:

V = π ∫ [f(x)]^2 dx By symmetry, we only need to consider the positive value of x From x^2 + y^2 = r^2 y^2 = r^2 - x^2 and the largest value of x is r So, V = 2π ∫ [f(x)]^2 dx [from 0 to r] = 2π ∫ (r^2 - x^2) dx [from 0 to r] = 2π(r^3 - x^3/3 | [0,r]) = 2π(r^3 - r^3/3) = 4πr^3/3

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