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Worked examples: interpreting definite integrals in context

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

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• Why would we use integrals to represent Julia's revenue? Isn't that unnecessarily complicated?
• I think integrals allow us to see the accumulation of her revenue at a certain interval of time.
• what would the definite integral of just k(t) from 0-4 be measuring? Its units would be kg*s.
• 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.
• 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 ?
• 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)
• 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.