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Unit 8

8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts

2 min readโ€ขjune 8, 2020

Anusha Tekumulla


๐ŸŽฅWatch: AP Calculus AB/BC - Interpreting the Meaning of a Derivative/Integral

Accumulation problems are word problems where the rate of change of a quantity is given and we are asked to calculate the value of the quantity accumulated over time. These problems are solved using definite integrals.

Resources:

What should you know about this topic? ๐Ÿง 

For this topic, youโ€™ll need to know how to do two things: interpret the meaning of a definite integral and determine the net change of an accumulation problem.ย 

In order to interpret the meaning of a definite integral, you must know two important things. Firstly, a function defined as an integral represents an accumulation of a rate of change. Second, the definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval.ย 

What will this look like on the AP Exam?ย โœ

In order to understand this concept, letโ€™s look at an example from an actual AP exam.ย 

2000 AB FRQ #4: In this question, water was being pumped into a tank at the constant rate of 8 gallons per minute and leaking out at the rate of โˆš(t+1) gallons per minute. At time t = 0 we are told there are 30 gallons of water in the tank.ย ๐Ÿฅ›

  • The first part of the question asked for the amount of water that leaked out of the tank in the first 3 minutes. To solve this, you must integrate the leak function from 0 to 3.ย 

  • The next part asked for the amount of water in the tank after t minutes. So we start with 30 gallons and add the amount put in which is 8 gallons per minute for 3 minutes of 24 gallons. Then we subtract the amount that leaked out from the first part. The amount is 30 + 24 โ€“ 14/3 gallons.ย 

  • The third part asked for an expression for A(t), the amount of water at any t. So following on the second part we have either

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