Pumping stations deliver gasoline at the rate modeled by the function D, given by d of t equals the quotient of 6 times t and the quantity 1 plus 2 times t with t measure in hours and and R(t)...

Question

Pumping stations deliver gasoline at the rate modeled by the function D, given by  with t measure in hours and and R(t) measured in gallons per hour. How much oil will the pumping stations deliver during the 3-hour period from t = 0 to t = 3? Give 3 decimal places.

Answer 1

Quantity = ∫Rate(t)dt for t = 0 to 3

Q = ∫³₀6t/(1+2t)dt

let u = 1+2t, du = 2dt, t = (u-1)/2

When t = 0, u = 1

when t = 3, u = 7

∫⁷₁6(u-1)/(4u)du = ∫⁷₁(3-3/(2u)) du = (3u - 3ln(u)/2|⁷₁ = (21-3ln(7)-3 + 0)/2 = 9- 3ln(7)/2 = 6.081 gallons