Short on time? This page is the recap. For explanations, the graphs and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- A rate of change graph shows how fast a quantity changes. The area between that graph and the horizontal axis over an interval is how much the quantity changed over the interval.
- Constant rate: change = rate × time, which is a rectangle’s area.
- If the graph is built from straight segments or circular arcs, find the area with geometry: rectangles, triangles, trapezoids, semicircles and quarter circles.
- Rate positive (graph above the axis): the quantity increases. Rate negative (graph below): it decreases.
- The area is a change, not an amount. Amount at the end = amount at the start + net change.
Key relationships
| Idea | Rule | Example |
|---|---|---|
| Units of the area | rate units × input units | (L/min) × min = L |
| Net change | area above − area below | 13.5 − 6 = 7.5 kWh |
| Total change (e.g. distance) | area above + area below | 2π + 18 ≈ 24.283 m |
| Amount at the end | start + net change | 4 + 7.5 = 11.5 kWh |
| Average rate over [a, b] | net change ÷ (b − a) | 2,450 ÷ 20 = 122.5 passengers/min |
Useful areas: triangle ½bh; trapezoid ½(a + b)h; semicircle ½πr²; quarter circle ¼πr².
Assumptions
- The graph shows a rate, not the quantity itself. Read the axis label before you start.
- Geometry gives an exact answer only when the pieces really are lines or circular arcs. Use the description and labelled points, not your ruler.
- “Area” in this topic always means area between the graph and the horizontal axis, over the stated interval.
Mistakes to avoid
- Giving the change as the final amount. Add the starting amount.
- Adding an area below the axis when the question asks for net change.
- Rate units on the answer, such as “litres per minute” for a volume.
- Counting grid squares without checking the scale on each axis.
- Interpretations without an interval or units.
Quick self-check
- Oil leaks from a machine at a constant 0.6 litres per hour for 5 hours. How much leaks out? (3 litres: the rectangle is 5 wide and 0.6 tall)
- A rate graph has area 7 above the axis and area 10 below it on [0, 9]. The quantity starts at 25. What is it at t = 9, and what is the total amount of change? (25 + 7 − 10 = 22; total change 7 + 10 = 17)
- A rate is measured in grams per minute and time in minutes. What unit does the area have? (grams)
Next: practice questions.