Solution SG 03

Consider a circle of radius 100-feet. Let the circle increase it’s radius by another 10-feet. What is the change in area? We provide two alternative solutions.
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  • Solution 1: A straight forward calculation that subtracts the final area from the initial area.
    • In[]:=
    Pi((100+10)^2-100^2)
    Out[]=
    2100π
    ◼
  • Solution 2:
    • In[]:=
    dA=D[Pir^2,r]
    Out[]=
    2πr
    ∴ dA = 2 π r dr
    Substituting values,
    In[]:=
    dA=2*Pi*100*10
    Out[]=
    2000π
    We have a problem that the two answers do not agree. Which is correct?
    Clearly Solution 1 is unquestionable. Solution 2 is incorrectly using r*dr to calculate the change. This is because dr is only an infinitesimal change operator which cannot be replaced by the change in radius of 10 ft.The correct way to use calculus to calculate dA is to evaluate the definite integral ∫dA; giving the total change in area. Whereby, ∫dA = ∫2 Pi r dR from 100 to 110.
    In[]:=
    dA=Integrate[D[Pir^2,r],{r,100,110}]
    Out[]=
    2100π