AP Calculus AB and BC · Exam guide
AP Calculus Rates and Accumulation FRQ: Tables, Riemann Sums, and Average Value
A rate-and-accumulation question gives a rate, often as a table, and asks what it adds up to. The integral of a rate is a total change, and the question is usually about units and meaning as much as arithmetic. This guide works one example from start to finish.
The big idea
If \(R(t)\) is a rate of change of a quantity, then \(\int_a^b R(t)\,dt\) is the net change in that quantity from \(t = a\) to \(t = b\). The amount at time \(b\) is the starting amount plus that net change.
A worked example
Water flows into a tank at a rate \(R(t)\) gallons per minute. Values of \(R\) are given in the table.
| t (minutes) | 0 | 2 | 5 | 8 |
|---|---|---|---|---|
| R(t) (gallons per minute) | 4 | 10 | 7 | 3 |
(a) Use a trapezoidal sum with the three subintervals from the table to approximate \(\int_0^8 R(t)\,dt\). Explain what it means. (b) Find the average rate of flow over \([0, 8]\), using your answer to (a).
(a) Trapezoids on \([0,2]\), \([2,5]\), \([5,8]\): \[\dfrac{2}{2}(4 + 10) + \dfrac{3}{2}(10 + 7) + \dfrac{3}{2}(7 + 3) = 14 + 25.5 + 15 = 54.5\] gallons. It approximates the total amount of water that flows into the tank in the first 8 minutes.
(b) The average value is \(\dfrac{1}{8}(54.5) = 6.8125\), which is about 6.813 gallons per minute.
What earns the points
| Part | Point | How to earn it |
|---|---|---|
| (a) | Setup | Show the trapezoid terms with the correct widths. Different widths are the usual source of errors. |
| (a) | Answer | A number with units (gallons). |
| (a) | Meaning | A sentence that says what the integral measures, in the context of the problem. |
| (b) | Answer | Divide the integral by the length of the interval, and give units of the rate. |
It describes a picture, not the quantity in the problem.
It uses the context and the units.
Left sums, right sums, and trapezoids
A left sum for the same table uses the left value on each subinterval: \(2(4) + 3(10) + 3(7) = 59\). Notice it is different from the trapezoid value of 54.5, because the rate rises and falls. When a question says an approximation is an overestimate or underestimate, justify it with whether the function is increasing, decreasing, or concave up or down on the interval.
Other things these questions ask
- How much is in the tank at a later time: the starting amount plus the integral.
- When the amount is greatest: where the rate changes from positive to negative, then check the endpoints.
- Whether a rate is increasing at a time: look at the sign of the derivative of the rate.
- The average value of the rate: the integral divided by the length of the interval.
Common mistakes
- Using the same width for every subinterval when the table spacing is uneven.
- Giving a number with no units.
- Treating a rate as an amount, or an amount as a rate.
- Forgetting the starting amount when asked for the amount at a time.
Frequently asked questions
Which approximation does the exam ask for?
Questions state which one to use: left, right, midpoint, or trapezoidal. Read the wording carefully.
Do I have to show the arithmetic?
Show the setup. A correct numerical answer with no setup may lose the setup point.
What is the average value of a function?
It is the integral of the function over the interval, divided by the length of the interval.
Do I need units?
Yes, whenever the answer is a quantity with units, and in sentences that explain meaning.
Where can I find the official scoring rules?
Exam formats change a little from year to year, so check the current course and exam description on the College Board site before test day.
Keep practicing
Every lesson in the AP Calculus courses ends with practice problems and an Exam focus box that shows how the topic is tested, and every unit has a 10-problem practice set with worked answers. These free lessons also build the skills the free-response section tests:
See the AP Calculus AB and BC course Free lessons and lesson list See the AP Calculus BC course
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Multiple choice
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