calculus propagated error question

Question

The measurement of the radius of the end of a log is found to be 26 inches, with a possible error of 1/2 inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log.

Answer 1

Let A(r) = area of the cross section of the end of the log = πr², r = radius of the end of the log

 

ΔA = actual  error in the measurement of the area 

 

 dA = A'(r)dr = approximate error   dr = error in measuring the radius (at most 1/2")

 

ΔA ≈ dA = A'(r)dr = 2πrdr = 2π(26")(±½")

 

                                       =  ± 26π in²