Derive the formula for the volume of a sphere?
1 Answer
# V = 4/3pi a^3#
Explanation:
Consider a 3-dimensional sphere of radius
In the
# x^2+y^2=a^2 #
Then if we take an arbitrary
# A = pir^2 #
# \ \ \ = pi(sqrt(a^2-x^2))^2 #
# \ \ \ = pi(a^2-x^2) #
We need to sum all of these infinitesimally thin vertical red slices as
# V = int_(-a)^(a) \ pi(a^2-x^2) \ dx #
# \ \ \ = pi \ int_(-a)^(a) \ a^2-x^2 \ dx #
# \ \ \ = pi \ [a^2x-x^3/3]_(-a)^(a)#
# \ \ \ = pi \ {(a^3-a^3/3) - (-a^3+a^3/3)}#
# \ \ \ = pi \ (a^3-a^3/3 +a^3-a^3/3)#
# \ \ \ = pi \ (2a^3-2/3a^3 )#
# \ \ \ = pi \ (4/3a^3)#
# \ \ \ = 4/3pi a^3#