# Differential Equations/Vol vs Area

Question
Hi Sir,

The volume of a sphere is 4/3 Pi r^3..the area is the differential 4Pir^2..why is this so..the first derivative is a slope..right? Thanks!!

It makes sense that dV/dr = surface area...

Think about it...if we increase the radius of a sphere by a small amount, say dr,
then the volume increases by about the surface area times the thickness of the
"extra layer" dr.  Hence, dV = 4*pi*r^2*dr ==> dV/dr=4*pi*r^2

See?

Abe

Differential Equations

Volunteer

#### Abe Mantell

##### Expertise

Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!

##### Experience

Over 15 years teaching at the college level.

Organizations
NCTM, NYSMATYC, AMATYC, MAA, NYSUT, AFT.

Education/Credentials
B.S. in Mathematics from Rensselaer Polytechnic Institute
M.S. (and A.B.D.) in Applied Mathematics from SUNY @ Stony Brook