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.
What should you know about this topic? 🧠
For this topic, you’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.
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.
What will this look like on the AP Exam? ✍
In order to understand this concept, let’s look at an example from an actual AP exam.
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 √(t+1) gallons per minute. At time t = 0 we are told there are 30 gallons of water in the tank. 🥛
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.
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 – 14/3 gallons.
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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