Video: Using Definite Integrals to Solve a Problem in Context

A tank is filled with water at a rate of π‘Ÿ(𝑑) = 𝑑² βˆ’ 𝑑 βˆ’ 6 cubic meters per minute. Find the amount of water that will be filled in the tank from 𝑑 = 2 to 𝑑 = 4 minutes. Round the answer to one decimal place.

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Video Transcript

A tank is filled with water at a rate of π‘Ÿ of 𝑑 equals 𝑑 squared minus 𝑑 minus six cubic meters per minute. Find the amount of water that will be filled in the tank from 𝑑 equals two to 𝑑 equals four minutes. Round the answer to one decimal place.

So, in order to solve this type of question, what we’re going to do is carry out integration. And that’s because what we’re trying to find is the amount of water that we filled in the tank between a time period of two minutes. So, what we can do here is form a definite integral. And the definite integral we have is the definite integral of 𝑑 squared minus 𝑑 minus six dπ‘₯. And that’s between limits two and four.

So, what we need to do first of all is remind ourselves what we do with definite integrals. Well, if we’ve got the definite integral of 𝑓 of π‘₯ dπ‘₯ that’s between our limits π‘Ž and 𝑏, then this is equal to the integral of 𝑓 of π‘₯ with 𝑏 substituted in for π‘₯ minus the integral of 𝑓 of π‘₯ with π‘Ž substituted in for π‘₯. So, the first stage is to integrate each of the terms in our function. So, what we’ve got is 𝑑 squared minus 𝑑 minus six. So therefore, if we integrate 𝑑 squared, we’re gonna get 𝑑 cubed over three. And just to remind us how we do that, what we do is we increase the exponent by one and divide by the new exponent.

So, it’s 𝑑 to the power of two plus one over to two plus one. And then, this is minus 𝑑 squared over two minus six 𝑑. So, now, what we need to do is we need to substitute in our limits two and four. So, when we do this, we’re gonna get four cubed over three minus four squared over two minus six multiplied by four. Then, minus two cubed over three minus two squared over two minus six multiplied by two. Which is gonna give us 64 over three minus eight minus 24 minus eight over three minus two minus 12. And what this is gonna be equal to is 0.6 recurring.

Well, the question asked for the answer to be to one decimal place. So, we’re gonna quickly round this. So therefore, we can say that the amount of water that will be filled in the tank from 𝑑 equals two to 𝑑 equals four minutes is going to be 0.7 cubic meters.

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