pls explain?

Question

an experimental car is found to have an efficiency of E(v),in miles per gallon of fuel,where v is the speed of the car.For a certain 4 hour trip, if v=v(t) is the speed of the car t hours after the trip started,what is the integral that represents the no. of gallons of fuel that the car used on the trip.

Answer 1

let x represent distance in miles
V represent volume in gallons
then
dx/dV = E is in miles/gallon
so
dV = 1/E dx
then
V
= ∫dV
= ∫ 1/E dx
= ∫ 1/E (dx/dt)dt
= ∫ (v / E) dt
since dx/dt = v

so V = ∫ v(t) / E(v(t)) dt in gallons

the end
.

Answer 2

EDITED ANSWER: Fuel efficiency is ∆s / ∆F, (miles per gallon), the inverse of what I had written. I replaced E(v) with 1 / E(v) for the correct answer.

Instantaneous efficiency is ∆s/∆F, where ∆F is the amount of fuel used in distance ∆s. In the limit, this becomes ds/dF. If you are given v(t), ds = v(t)*dt so

E[v(t)] = v(t)*dt / dF

dF = v(t)*dt / E[v(t)]

Total fuel used is then ∫[t=0 to T] dF = ∫[t=0 to T] v(t)/E[(v(t)] * dt