Compute the volume of a sphere of radius R Use the shell met

Solution

Volume = 2Integrate(a , b) x*g(x).dx

Sphere x^2 + y^2 + z^2 = r^2

Sphere in XY plane is a circle (x^2+y^2 = Constant)

we can also further come into Axis

x^2=y^2=z^2 = Constant

But Due to Symetry about y axis

2x^2=2y^2=2z^2 = Constant

So we can let g(x) = 2x^2

But Due to Symetry about y axis

Therefore

Volume = 2Integrate(R , 0) x*2x.dx

= (4/3)(x^3)₀^R

= (4/3)R^3