Finding the total power delivered

I'm really only concerned with the setup so I'm just going to ask it in a general sense, rather than the specific problem.

Homework Statement

Suppose you have a constant voltage and the current is described by i(t). Find the total power delivered between time A and time B.

The Attempt at a Solution

I got in a discussion with my classmates, and we couldn't settle which way is correct. So P(t) = V*i(t). Would it be set up as ∫ dP from A to B, or ∫ P(t) dt from A to B?

The first is summing small bits of power and will return a unit in Watts. The second sums energy and returns a unit in Joules. So which setup is right, and would the total power delivered end up being power, or energy?

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gneill
Mentor
I'm not familiar with any definition of "total power delivered". Average power, sure, but not total power. Power is a rate of energy production or consumption with the units of watts or Joules per second.

The question would make more sense if it asked for total energy delivered over the given interval, or even average power delivered over that time.

When you set up a definite integral, the endpoints must have the same units as the differential elements being summed. So ∫dP would require end limits that are given in units of watts, not time.

So, I guess I'm saying that I'm not sure what to suggest given that the problem in its stated form doesn't make sense to me Is there another statement of the problem, perhaps in a context that would make it clearer?

Well this is the full question:

The voltage v is a constant 10 volts and the current i is described by the following function:

i(t) = (5t^2+20+6)/(t^3+2t^2+t) Amps

What is the total power delivered between t=1s and t=5s?

I figured since P is a function of time it would work since ∫ dP = ∫ P'(t) dt. And that would allow units in time. Or you could do it ∫ dP from P(A) to P(B), that would be the same.
It was also the phrase "total power delivered" that threw everyone else in my class off.

Last edited:
gneill
Mentor
Differentiating the function just to integrate it again doesn't strike me as being a particularly useful exercise.

I suspect that it was intended that the question ask you find the total energy delivered over the time interval rather than the total power, but was hit by a typo on the way to school

That makes sense. It struck me as odd that that was the way to solve it, but it was asking for power, and I couldn't figure out any other way to get a unit of power. Thanks.