Find the amount of oil that the field can be expected to yield during the first 6 years of production, assuming that the projection holds true.

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)=((700t^2)/(t^3+32))+5 (0 ≤ t ≤ 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 6 years of production, assuming that the projection holds true. (Round your answer to the nearest thousand barrels.)

Answer 1

I believe if you integrate R(t) in the interval from 0 to 6 years you will obtain the number of barrels produced in 6 years. So let’s start the integration:

∫((700t²)/(t³ + 32) + 5,t,0,6)

*the integral notation used is what a TI-89 or better uses to integrate.

I will integrate (700t²)/(t³ + 32) by using substitution. Let u = t³ + 32 and du = 3t²dt. The substitution will change the first part of the integral ∫(700t²)/(t³ + 32) dt into: ∫(700)/(3u) du = 700/3 ln u

Reversing the substitution yields:

700/3ln(t³ + 32) + 5t which needs to be evaluated in the interval [0,6]. Evaluation on this interval will result in the following expression:

700/3ln((6³ + 32)/(0³ + 32)) + 5(6)

700/3ln(248/32) + 30

700/3(2.048) + 30 ≈ 508 thousand barrels