# Summation Solution differing from Integral Solution

I know this is elementary stuff but I'm hoping to clear get it cleared up.

I have a money making machine. I turn on the machine and for the first day I make $5.70. For every subsequent day, I make a profit that is$.35 less.

I want to use basic calculus to find how much I've made in (t) days.

So from what I can see, my profit on a given t(the day, 1st, 2nd, 3rd, etc) is given by

profitPerDay = 5.70 - .35(t)


If I want the profit after 3 days I take the following definite integral from t=0, to t=3

Integral(t=0, t=3) [5.70 - .35(t)]dt


and I get

profit(day) = 5.7t - .35t^2 = 14.65


But when I simply sum up the 3 profits over each day given by

profitPerDay = 5.70 - .35(t)


and compare, I get 15.00 instead

[5.7 - .35(1)] + [5.7 - .35(2)] + [5.7 - .35(3)]  = 15.00


What am I doing wrong in this integral?

The integral sums continuously whereas the normal sum is discrete. What you have done is integrated the expression $$5.7-.35t$$, and approximated the same integral with a Riemann sum. The picture below shows how the approximation works
• The way the machine is described in the question, it seems that, even if it outputs money continuously, the right answer is given by the sum, not the integral. The reason is that the question says "for the first day I make 5.70", not that the machine is initially outputting money at a rate of 5.70 per day which could then decrease gradually during the first day. The numbers in the question are already integrals over $1$-day periods. Commented Apr 27, 2013 at 20:01