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Posted by: Gman5434
Some one prove that V=(4/3)pi*r^3 for a sphere using Calculus! Consider a sphere of radius R centered at the origin, described by the equation x^2 + y^2 + z^2 = R^2. We wish to find its volume.
An appropriate coordinate transformation would be:
x = r*sin(p)cos(t)
y = r*sin(p)(sin(t)
z = r*sin(p)
With domain D = {0 <= r <= R, 0 <= t <= 2pi, 0 <= p <= pi}
The determinant of the Jacobian matrix for this transformation is r^2*sin(p)
The volume of the sphere is then given by the triple integral over D of
r^2*sin(p)drdpdt.
Since the terms are independent, the volume integral can be expressed as the product of simple integrals, resulting in the following:
Integral of r^2 from r = 0 to R
Integral of sin(p) from p = 0 to pi
Integral of dt from t = 0 to 2pi
Which evaluates to
(1/3)r^3|0,R --> (R^3)/3 - 0 = (1/3)R^3
-cos(p)|0,pi --> -(-1) - (-1) = 2
t|0,2pi --> 2pi
Multiplying the results yields V = (4/3)*pi*R^3