In

http:#33329334 you state a problem where you are looking for the total population after 15 years (not the increase in population).

dp(t)/dx = (4+0.15t)^(3/2)

dp(t) = (4+0.15t)^(3/2) dx

Integrating from 0..15:

Delta p = p(15) - p(0) = 175.08

So, 175.08K represents the increase in the frog population in 15 years.

p(0) = 100K (initial condition)

p(15) = 175.08 + p(0) = 275.08K = total frog population in year 2015

Since the book said "Estimate the population", I interpret that to mean "Estimate the total population ". If that is the book's interpretation, then the book's answer of 175K is wrong. If the book's author interpreted it instead as "Estimate the increase in population", then the book's answer is right.

I hope you never have to take a multiple choice test written by the book's author.

The good news for you is that with so many ambiguities in the book, this really makes you think hard about what you are modeling; and this is likely to give you a deeper understanding of word problems.

Maybe the author plugged in the formulas real quick into an integrator and copied the 175 at t=15 forgetting that the left hand side of the integral yields Delta P = P(15) - P(0).