Area under a curve

The definite integral is the exact area between a curve and the x-axis. Rectangles or trapezia estimate it. Add more strips, change the method and the limits, and watch the estimate close in on the exact value.

Predict first

Start from the defaults (f(x) = 0.3x² + 0.5, a = 1, b = 4, n = 8, midpoint rectangles). If you switch the method to Left rectangles, how does the estimate compare with the exact integral?

curve and strips estimate (this method) exact value

Function

Method

Strips, n

Lower limit, a

Upper limit, b

Preset

exact = F(b) − F(a)
width h = (b − a) ÷ n
left: h × Σ f(xᵢ), right: h × Σ f(xᵢ₊₁)
midpoint: h × Σ f(xᵢ + h/2)
trapezium: (h ÷ 2) × [f(x₀) + 2f(x₁) + … + f(xₙ)]

Estimate:
Exact integral:
Error (estimate − exact):
Relative error:
Strip width h:

Try this

Use f(x) = 0.3x² + 0.5 from a = 1 to b = 4 with the left and then the right rectangle method at n = 8. Which one under-estimates and which over-estimates, and why?
Keep the trapezium rule and double n from 5 to 10. Roughly how does the error change? Compare with the midpoint method.
Choose f(x) = sin(x) + 1.2 and the limits a = 0, b = 5. The curve is concave on part of the interval, so look at where the trapezium rule over-estimates and where it under-estimates.

Exam tip: for an increasing function the left rectangles give an under-estimate and the right rectangles an over-estimate, so the exact area lies between them; for a concave-up curve the trapezium rule over-estimates, for concave-down it under-estimates.