Estimate the area under a curve by splitting it into trapezia, say whether the estimate is too big or too small, and interpret the area as a distance or a total amount.
What you will learn
- Estimate the area under a curve with strips of equal width, using trapezia
- Say whether a trapezium estimate is an overestimate or an underestimate from the shape of the curve
- Interpret the area under a speed-time graph as a distance, and under a rate graph as a total
- Work out an exact distance from a speed-time graph made of straight lines, and an average speed
Strips and trapezia
Split the area into strips of equal width h.
Join the tops of each strip with a straight line, so each strip is a trapezium with area ½ × h × (a + b), where a and b are the heights at its two edges. Add the trapezia.
A car's speed is read every 2 seconds:
- speeds at t = 0, 2, 4, 6, 8: 0, 6, 10, 12, 12
- ½ × 2 × (0 + 6) = 6
- ½ × 2 × (6 + 10) = 16
- ½ × 2 × (10 + 12) = 22
- ½ × 2 × (12 + 12) = 24
- distance ≈ 6 + 16 + 22 + 24 = 68 m
The quick form: ½ × h × (first + last + 2 × the middle heights) = ½ × 2 × (0 + 12 + 2 × 28) = 68.
The rest of this lesson and its 8 practice questions are locked.
In this lesson
- Strips and trapezia
- Too big or too small?
- What the area means
- Straight-line graphs give exact areas
- Try it: First question
8 practice questions, marked as you go; about 30 minutes in all.
Key terms
Above, Underestimate, Below, Overestimate, Percentage error, Distance, Volume
In this unit: Graphs and coordinates: shapes on axes, regions, areas and tangents
More graphs and coordinate geometry: shapes on coordinate axes, gradients and perpendicular lines in shapes, regions with strict and inclusive boundaries, the area under a curve from trapezia, and the gradient at a point from a tangent.
- Shapes on coordinate axes
- Gradients and lines in shapes
- Regions in depth
- Area under a curve
- Gradient at a point, and the kinematics formulae
All of Graphs and coordinates: shapes on axes, regions, areas and tangents