If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Worked examples: interpreting definite integrals in context

Interpreting expressions involving definite integrals in a real-world context.

Want to join the conversation?

  • orange juice squid orange style avatar for user Tyler
    Why would we use integrals to represent Julia's revenue? Isn't that unnecessarily complicated?
    (12 votes)
    Default Khan Academy avatar avatar for user
  • piceratops ultimate style avatar for user Joshua
    what would the definite integral of just k(t) from 0-4 be measuring? Its units would be kg*s.
    (5 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user mshelikoff
      The owner of the sauce factory might charge a potential competitor who doesn't have a factory—perhaps because they are just starting out in the business—a usage fee of $1 per hour per kilogram of ketchup produced. The definite integral of k(t) from 0 to 4 would then measure the total fee for those four hours.
      (5 votes)
  • blobby green style avatar for user mohzakiyah1997
    For the ketchap problem, what happens if we take the integral to K(t) rather than K'(t). We will get something Kg hours. What does this means ?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • starky ultimate style avatar for user Ray2017
    Are the domains of k'(t) discrete in the second problem? Since t is in hours, would t have to be integers only or does it take any real number as its domain?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • male robot hal style avatar for user KEVIN
    Wouldn't the 3 represent the amount made from time = 0 to time = 1, if we put time as the horizontal axis? The month has to elapse; it doesn't just exist. The 3 and the integral from 1 to 5 are separate values/entities, yes? Could we say that 3 = the definite integral from 0 to 1 r(t) dt and we add to that the definite integral from 1 to 5 r(t) dt? We then add those together and get our $19,000. Any context or perspective is appreciated.
    (1 vote)
    Default Khan Academy avatar avatar for user
    • male robot donald style avatar for user Venkata
      3 is indeed the amount of money made from t = 0 to t = 1. Essentially, t = 0 is when no months have passed, and t = 1 is when 1 month has passed. So, going from t = 0 to t = 1 means one month has passed.

      We could write 3 as the integral of r(t) from 0 to 1, but here's the issue: we don't know if the rate of her getting revenue between t = 0 to t = 1 is the same as the rate between t = 1 and t = 5. So, with the information given, we can't write 3 as an integral.

      Hope that made sense?
      (2 votes)

Video transcript

- [Instructor] Julia's revenue is r of t thousand dollars per month, where t is the month of the year. Julia had made $3,000 in the first month of the year, what does three plus the definite role from one to five of r of t dt equals 19 mean? And we have some choices. So like always, pause the video and see if you can work through it. Alright, now let's work through this together. So they tell us that she made $3,000 in the first month and we also see this three here, so that's interesting. Maybe they represent the same thing, we don't know for sure yet. But let's look at this definite integral. The definite integral from one to five of r of t dt, this is the area under this rate curve, r of t is the rate atwhich Julia makes revenue on a monthly basis. So if you take the area under that rate curve, that's going to give you the net change in revenue from month one to month five, how much that increased. And so if you add that to the amount she made in month one, well that tells you the total she makes from essentially time zero all the way to month five. And they're saying that is equal to 19. So let's see which of these choices are consistent with that. Julia made an additional $19,000 between months one and five. Choice A would be correct if you didn't see this three over here. Because just the definite integral is the additional between months one and five, but that's not what this expression says, it says three plus this is equal to 19. If it said Julia made an additional $16,000, well that would make sense because you could subtract three from both sides and you'd get that result, but that's not what they're saying. Julia made an average of $19,000 per month. Well once again, that's also not right because we just said from the beginning, from time zero, all the way until the fifth month, she made a total of $19,000. Not the average per month is $19,000. Julia made $19,000 in the fifth month. Once again, this is not just saying what happened in the fifth month. This is saying, we have the $3,000 from the first month and then we have the additional from month between month one and month five, so that's not that. So this better be our choice. By the end of the fifth month, Julia had made a total of $19,000. Yes that is correct. She made $3,000 in month one and then as we go between month one to the end of month five, to the end of the fifth month, she has made a total of $19,000. Let's do another one of these. So here we're told the function k of t gives the amount of ketchup in kilograms produced in a sauce factory by time in hours on a given day. So this is really quantity is a function of time, it isn't rate, what does the definite integral from zero to four of k prime of t dt represent? Once again, pause the video and see if you can work through it. Well k of t is the amount of ketchup as function of time. So k prime of t, that's going to be the rate at which our amount of ketchup is changing a function of time. But once again, when you're taking the area under the rate curve, that tells you the net change in the original quantity in the amount of ketchup. And as the net change between time zero and time four, so let's see which of these choices match up to that. The average rate of change of the ketchup production over the first four hours. No, that does not tell us the average rate of change, there's other ways to calculate that the time it takes to produce four kilograms of ketchup, so does this represent the time it takes? To represent four kilograms of ketchup. No, this four is a time right over here. This is gonna tell you how much ketchup gets produced from time zero to time four. The instantaneous rate of production at t equals four. Now this would be k prime of four, that's not what this integral represents. The amount of ketchup produced over the first four hours, yep that is exactly right.