Graphs
Grade 8-9Area Under a Graph & Rate of Change
The area under a graph and the gradient of a graph both carry real meaning depending on what the axes represent, whether that is total distance travelled or the speed at a single instant. This lesson covers using the trapezium rule to estimate area under a graph, finding the gradient of a curve at a point using a tangent, and interpreting what gradient and area represent, including their units.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Estimating area under a graph with the trapezium rule
The trapezium rule estimates the area under a graph by splitting it into equal-width strips and treating each strip as a trapezium. With strip width h and y-values y₀, y₁, ..., yₙ, the area is estimated as (h/2) × [(first value + last value) + 2 × (sum of the remaining middle values)].
For a speed-time table with strip width 2 and values 0, 3, 5, 4, 2: area ≈ (2/2) × [(0 + 2) + 2(3 + 5 + 4)] = 1 × [2 + 24] = 26.
Finding the gradient of a curve at a point
The gradient of a curve at a single point is found by drawing a tangent, a straight line that just touches the curve at that point, and calculating the gradient of that tangent using two points on it.
If a tangent to a curve at point P passes through (1, 0) and (4, 12), the gradient at P is (12 - 0) ÷ (4 - 1) = 4.
Interpreting the area under a graph
The area between a graph and the horizontal axis represents the product of the two quantities on the axes. On a speed-time graph, since speed × time = distance, the area under the graph represents the total distance travelled.
The area under a speed-time graph, measured in metres per second multiplied by seconds, represents the total distance travelled, in metres.
Interpreting the gradient of a graph
The gradient of a graph represents the rate of change of the quantity on the vertical axis with respect to the quantity on the horizontal axis, and its units combine the units of both axes. Comparing gradient values at two points shows whether that rate is increasing or decreasing.
The gradient of a volume-time graph represents the rate at which the tank is being filled, in litres per minute. If the gradient increases from 2 m/s at point A to 7 m/s at point B on a distance-time graph, the speed is increasing between A and B.
Worked Examples
Three exam-style questions, fully solved.
The graph shows a curve with point P marked on it. The tangent to the curve at P passes through the points (1, 0) and (4, 12). Estimate the gradient of the curve at P.
Easy- 1.Find the change in y between the two points on the tangent: 12 - 0 = 12
- 2.Find the change in x between the two points on the tangent: 4 - 1 = 3
- 3.Divide the change in y by the change in x: 12 ÷ 3
Answer: 4
The table shows the speed, v (m/s), of a cyclist at time t (seconds). Use the trapezium rule with 4 strips to estimate the total distance travelled between t = 0 and t = 8.
Medium- 1.Identify the strip width: h = 2
- 2.Add the first and last values: 0 + 2 = 2
- 3.Double the sum of the middle values: 2 × (3 + 5 + 4) = 24
- 4.Apply the formula: (2 ÷ 2) × (2 + 24)
Answer: 26 m
A graph shows the volume of water in a tank, V (litres), against time, t (minutes), as the tank is filled. State what the gradient of the graph represents, including its units.
Hard- 1.Recognise that the gradient is the rate of change of volume with respect to time
- 2.Combine the units of the two axes: litres and minutes
Answer: The rate at which the tank is being filled, in litres per minute
Avoid These
The most common mistakes students make.
Forgetting to multiply by h ÷ 2 (half the strip width) when applying the trapezium rule.
Forgetting to double the middle values in the trapezium rule formula, treating them the same as the first and last values.
Estimating a curve's gradient using two points that lie on the curve itself, instead of two points on the tangent line drawn at that point.
Giving a gradient or area value without stating what it represents in context, or leaving out the correct units.
Applying the "area represents distance" rule to a distance-time graph, where the area has no standard physical meaning, instead of a speed-time or velocity-time graph.
FAQ
Questions parents and students ask.
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