AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/6/6-1)
Unit 6 · Topic 6.1
6.1 Exploring Accumulations of Change
When you know how fast something is changing, the area between the rate graph and the horizontal axis tells you how much it changed in total. This idea, accumulation, is the starting point for everything in the rest of the course about integrals.
Key terms
- accumulation
- rate of change
- net change
- area under a curve
- units
Area under a rate graph is a change in amount
Say water flows into a tank at a steady 4 gallons per minute for 5 minutes. You'd say 4 × 5 = 20 gallons went in. On a graph of rate against time, that's a horizontal line at height 4, and the rectangle under it from t = 0 to t = 5 has area 4 × 5 = 20. The area is the amount.
That works even when the rate isn't constant. Chop the time into tiny pieces. On each tiny piece the rate is almost constant, so rate × time is almost exactly the amount that came in, and that product is the area of a thin strip. Add all the strips and you get the area under the curve. So for any rate graph, the area between the graph and the time axis equals the accumulated change: the total amount added or removed over that time.
Area below the axis counts as negative
A rate can be negative. If r(t) is the rate water flows into a tank and r(t) is negative, water is draining out. Area that sits below the t-axis counts as negative change. Area above the axis counts as positive change.
The net change over an interval is the area above the axis minus the area below it. If the two are equal, the amount ends where it started, even though plenty happened along the way.
- Rate positive on an interval → the amount goes up over that interval.
- Rate negative on an interval → the amount goes down over that interval.
- Rate changes from positive to negative → the amount reaches a local high point there.
- Rate changes from negative to positive → the amount reaches a local low point there.
Units of the area
The units of the area are the rate's units times the horizontal axis's units. Gallons per minute × minutes = gallons. Meters per second × seconds = meters. People per hour × hours = people.
Checking units is a fast way to see what an area means. If the area comes out in “meters,” it's a change in position. If it comes out in “gallons,” it's a change in volume. Free-response questions often ask you to explain what an area or integral means, and the units are part of a full answer.
Finding the area with geometry
In this topic the rate graphs are usually made of straight segments and sometimes semicircles, so you can find areas with shapes you already know: rectangles (base × height), triangles (½ × base × height), trapezoids (½ × (b₁ + b₂) × h) and semicircles (½πr²). Split the region where the graph crosses the axis, find each area, and give each one a sign.
Later (6.3 and 6.7) you'll write this area as a definite integral, ∫ₐᵇ r(t) dt, and compute it without pictures. The meaning stays the same: the integral of a rate is the net change in the amount.
| Rate (vertical axis) | Horizontal axis | Area means |
|---|---|---|
| velocity (m/s) | time (s) | change in position, or displacement (m) |
| water flow (gal/min) | time (min) | change in volume (gal) |
| birth rate (people/yr) | time (yr) | number of people added |
| marginal cost (dollars per item) | items | change in cost (dollars) |
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Net change from a rate graph
Water flows into a tank at rate r(t) liters per hour, for 0 ≤ t ≤ 8. The graph of r is made of segments: r(t) = 3 on 0 ≤ t ≤ 2; a straight line from (2, 3) down to (6, −3), crossing the axis at t = 4; and r(t) = −3 on 6 ≤ t ≤ 8. The tank holds 50 liters at t = 0. (a) How much water is in the tank at t = 8? (b) At what time is the amount greatest, and what is it?
Show the solutionHide the solution
- Step 1: Split the region where the graph crosses the axis (t = 4), and find each area with geometry.
- Step 2: From 0 to 2: rectangle, 2 × 3 = 6 liters, above the axis, so +6.
- Step 3: From 2 to 4: triangle with base 2 and height 3, ½(2)(3) = 3, above the axis, so +3.
- Step 4: From 4 to 6: triangle with base 2 and height 3, below the axis, so −3.
- Step 5: From 6 to 8: rectangle 2 × 3 below the axis, so −6.
- Step 6: (a) Net change = 6 + 3 − 3 − 6 = 0 liters. Amount at t = 8 is 50 + 0 = 50 liters.
- Step 7: (b) The amount rises while r > 0 (0 < t < 4) and falls while r < 0 (4 < t < 8). So the most water is at t = 4, when r changes from positive to negative. Amount = 50 + 6 + 3 = 59 liters.
Answer: (a) 50 liters. (b) The maximum is 59 liters, at t = 4 hours.
- Example 2
Trap: saying what the area means
Cars enter a parking garage at a rate of c(t) cars per hour, where t is hours after 6 a.m. The area under the graph of c from t = 2 to t = 5 is 340. Explain what 340 means.
Show the solutionHide the solution
- Step 1: Units first: (cars per hour) × (hours) = cars.
- Step 2: The interval t = 2 to t = 5 is 8 a.m. to 11 a.m.
- Step 3: A weak answer says “340 is the area under the curve.” That's true but earns nothing in context.
- Step 4: A full answer names the quantity, the units and the time interval: the total number of cars that entered between 8 a.m. and 11 a.m.
Answer: 340 cars entered the garage between 8 a.m. (t = 2) and 11 a.m. (t = 5).
Common mistakes
- Adding all the areas as positive when you want net change. Area below the axis subtracts. Only add everything as positive when the question asks for total amount moved, like total distance.
- Forgetting the starting amount. Area gives the change, so the final amount is the starting amount plus the net change.
- Getting the units wrong, such as writing gallons per minute for an area. Multiply the rate's units by the horizontal axis's units.
- Thinking the amount is greatest where the rate is greatest. The amount is greatest where the rate switches from positive to negative.
On the exam
- Free-response questions often give a rate graph made of segments and semicircles and ask for an amount at a later time. Show each geometric area with its sign, then add the starting value.
- When asked to interpret an area or integral, include the quantity, the units and the time interval in one sentence.
Connected topics
Videos
Check yourself
4 questions on 6.1 Exploring Accumulations of Change. Pick an answer to see if you got it, and why.
Rain falls on a town at a rate modeled by r(t) = 0.3 + 0.2 sin(t²/4) inches per hour, where t is the number of hours since the rain began, for 0 ≤ t ≤ 4. To the nearest thousandth of an inch, how much rain falls during these 4 hours?
E(t) is the electricity demand in a city, in megawatts, where t is measured in hours after midnight. It is known that ∫ from 14 to 22 of E′(t) dt = −40. Which of the following is the best interpretation of this fact?
The temperature of a liquid in a lab changes at a rate of T′(t) = 3cos(t/2) degrees Celsius per minute, where t is measured in minutes for 0 ≤ t ≤ 8. At time t = 0, the temperature of the liquid is 40°C. What is the temperature of the liquid at time t = 8?
| t (minutes) | R(t) (gallons per minute) |
|---|---|
| 0 | 4 |
| 2 | 7 |
| 5 | 9 |
| 7 | 10 |
| 12 | 11 |
Invented data: rate of water flowing into a tank
R(t) is the rate, in gallons per minute, at which water flows into the tank at time t minutes. Which of the following best describes the meaning of ∫₀¹² R(t) dt in this context?
0 of 4 answered