ok, so there's no way that answer can be right. that's less pens than they are currently producing and they tell us that the rate is increasing, we should expect at least 20000 at the end of 5 years. doing it by hand without calculus the answer comes up to around 24400

and even if you did that integral, the answer would be close to 50000

i would probably model the derivative as a line, since now when t=0, we produce 4000, that would be our y-intercept, since we are increasing at a rate of 10%, 0.1 would be our slope, so our function is:

P = 0.1t + 4000

int{P} = 0.05t^2 + 4000t

evaluated between 0 and 5 we get 200001.25, closer to our expected answer but not quite. i haven't really figured out what's causing the lack of that 400 units yet