Example 2

If money is received at the rate f(t) = 2t dollars per year, and earns interest at the annual rate of 7%, how much will be accumulated from times r = 0 to f = 10?

The formula gives

We first find the indefinite integral.

Let u = -0.07t, du = -0.07 dt. Then

Using integration by parts,

Therefore

When f = 0, u = 0 and when f = 10, u = -0.7. Thus

The answer is $ 128.08.

Notice that if the money were placed under a mattress and earned no interest, the capital accumulated between times f = 0 and t = 10 would be

The formula for capital accumulation also has a meaning when f(t) is negative part or all of the time. A negative value of f(t) means that money is being paid out instead of received. When f(t) is negative, money must be either withdrawn from the bank account or else borrowed from the bank at interest rate r. The formula

then represents the net gain or loss of capital from times f = a to f = b, provided that the bank pays interest on savings and charges interest on loans at the same rate r.