Definite Integrals and Substitution

Question

Based on a preliminary report by a geological survey team, it is estimated that a newly discovered oil field can be expected to produce oil at the rate of

R(t)= (800t^2 / t^3+31)+5 (0 less than or equal to t less than or equal to 20

thousand barrels/year, t years after production begins. Find the amount of oil that the field can be expected to yield during the first 7 years of production, assuming that the projection holds true. (Round your answer to the nearest thousand barrels.)

Answer 1

Hello, Brendon!

The first thing to notice is that the equation given is for the rate of oil production. So to get the amount of oil, we need to take the integral.

∫ (800t² / (t³+31)) + 5 dt

First I'm going to separate the 5 to make this slightly easier later:

∫ (800t² / (t³+31)) dt + ∫ 5 dt

I'm going to use u substitution and make u = t³+31

Therefore, du= 3t² dt

So now we have ∫ (800t²/u) dt + ∫ 5 dt

We can factor a 3 out of the 800 to turn our dt into du, leaving us with:∫

266.67 * ∫ 1/u du + ∫ 5 dt

Now, our bounds for the dt side are as given in the original problem: 0 and 7. If we plug those values in for t in the equation u = t³+31, we get a lower bound of 31 and an upper bound of 374.

The integral of 1/u is ln|u|, so we need to evaluate that from 31 to 374 then multiply the answer by 266.67. Doing that, we get 664.07

We have an easier integral on the dt side--just evaluating 5t from 0 to 7. Doing that, we get 35

Adding those two, we get 664.07+35 = 699.07. Rounding, that's 699,000 barrels in the first 7 years.

Hope that helped!