Area under graphs

Objective. Estimate accumulated quantities using graphical areas.

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Learn: the key idea

Area under graphs is estimated by dividing the region between a curve and the x-axis into shapes, typically trapezia of equal width, calculating the area of each one, and summing them to approximate quantities like total distance from a speed–time graph. Each trapezium's area is found using ½(a+b)h, where a and b are the parallel sides (the function values at each end) and h is the width of the strip. Using more, narrower strips generally increases the accuracy of the estimate since the trapezia follow the curve more closely. For example, estimating the area under a speed-time graph gives the total distance travelled, even when the speed varies continuously.

Worked example

Estimate the area under a curve between x=0 and x=4 using two trapezia of width 2, given y-values of 3, 7 and 5 at x=0, 2 and 4.

1 $First trapezium (x=0 to 2): area = ½(3+7)\times2 = 10$
2 $Second trapezium (x=2 to 4): area = ½(7+5)\times2 = 12$
3 $Add both areas together: 10+12$
4 $Total estimated area = 22$

Answer: 22 square units

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Common mistake. Students sometimes use the rectangle rule with only the starting height of each strip, which can noticeably underestimate or overestimate curved areas; the trapezium rule using both parallel sides usually gives a closer estimate to the true area.

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