Integration, Integration by parts, and present value

Part 1.

The projected fuel cost C(in millions of dollars per year) for a trucking company from 2008 through 2020 is C1 = 5.6 + 2.21t, 8<= t<= 20, where t = 8 corresponds to 2008. If the company purchases more efficient truck engines, fuel cost is expected to decrease and to follow the model C2=4.7+2.04t, 8<=t<=20. How much can the company save with the more efficient engines?

Part 2.

Assuming a 10 percent discount rate, what is the most you would pay to purchase the more efficient truck engines? Taking into consideration of the inflation rate in the 12 years period. And use this present value formula.

Using this formula of Present Value:

If c represents a continuous income function in dollars per year and the annual rate of inflation is r, then the actual total income over t1 years is

t1

Actual income over t1 year = [ c(t) dt

0

And its present value is

t1

Present Value = [ c(t)e^ -rt dt

0

[ this is a definite integral sign, sorry I couldn't get this sign. I use the bracket for now to represent it.

I used the definite integral to get 39.4 millions the company save if they choose to use the efficient engines. Now I don't know how the get the function to find the current value. Help asap, problem due monday.