What it shows
The area under a curve can be approximated by many thin strips. The more strips used, the closer the sum is to the definite integral.
The science
For y = x squared from 0 to 4 the exact area is the integral of x squared, which is x cubed / 3 from 0 to 4, or 64/3, about 21.33. Strips of width h whose heights are taken at their left edge give a lower sum for a rising curve, and the right edges give an upper sum. The exact area lies between them. The trapezium rule averages the two and is much closer. As the strips narrow, both sums close in on the integral, which is the idea behind Riemann sums.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- Graph paper with 1 mm squares
- A pencil and ruler
- Coloured pencils
- A calculator
- A table of values for y = x squared
- Scissors and glue for a cut-out comparison
- A thin card to cut out the shape
- A scale reading to 0.01 g (a jeweller’s shop will weigh the card pieces)
Safety
For this project: An adult should supervise the use of scissors for younger students.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
Step by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Draw the curve y = x squared from x = 0 to 4 on graph paper, with 1 cm for each unit of x and 1 cm for 2 units of y.
Divide the area into 4 strips of width 1. Draw rectangles using the left-hand heights and shade them. Add their areas for the lower sum.
Draw the upper rectangles using the right-hand heights in another colour and add them for the upper sum.
Repeat with 8 strips of width 0.5, using a table of values for y at 0, 0.5, 1 and so on.
Calculate the trapezium estimate for 4 strips: the width times the sum of half the first and last heights and all the middle heights.
Count the squares under the curve on the graph paper to get a measured area and compare with the sums.
Cut out the region under the curve from thick card, weigh it, and compare with the weight of a 4 by 16 rectangle. The ratio of weights gives the area as a fraction of 64.
Make a table of lower sum, upper sum, trapezium sum, counted squares and the exact 21.33 for each method with the percentage error.
What you should see
The lower sum is under 21.33 and the upper sum is over it, the trapezium estimate is within about 3 per cent with 4 strips and under 1 per cent with 8, and doubling the number of strips halves the gap between the lower and upper sums.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
01What is the exact area under y = x squared from 0 to 4?
64/3, about 21.33 square units.
02Why is the left-hand sum a lower estimate here?
The curve is rising, so each left height is smaller than the heights across the strip.
03Why is the trapezium rule more accurate?
It follows the slope of the curve by joining the two heights with a straight line.
04What is a definite integral?
The limit of such sums as the strips become infinitely thin.
Take it further
Repeat for y = sin x from x = 0 to x = pi radians (180 degrees) and see that the area is exactly 2.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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