Dynamic Calculus
Home > Integral calculus
Trapezoidal rule
When the primitive function is known, the exact area under a curve can be found by means of the definite integral. For functions whose definite integrals cannot be easily found, numerical methods can be used to approximate the value of the required area. One such method, the trapezoidal rule, divides the area into a given number of trapezia with equal widths and makes use of the formula for the area of a trapezium.
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
- Calculate the exact area bounded by the curve \(f(x) = 0.1{x^2} + 4\), the \(x\)-axis and the ordinates \(x = 1\) and \(x = 8\). Check your answer by clicking the 'checkbox'.
- Confirm by calculation that \(f(a) = f(1) = 4.1\) and \(f(b) = f(8) = 10.4\)
- Using the formula for the area of a trapezium, approximate the area under the curve with one trapezium (\(n = 1\)).
- Move the 'slider' to position \(n = 2\). Given that \(f(a)\) is the length of the shorter of the parallel sides of the left trapezium and \(f(b)\) is the length of the longer of the parallel sides of the right trapezium, what is the length of the other parallel side? Express the height of the trapezia in terms of \(a\) and \(b\).
- Using your answers from (4), write down an expression for the sum of the areas of the two trapezia. Is your answer the same as the formula shown in the applet for \(n = 2\)?
- The width of each sub-interval (or height of each trapezium) is given by \(h = \frac{{b - a}}{n}\), where \(n\) = number of sub-intervals. The \(x\)-values of the ordinates are then \(a\), \(a + h\), \(a + 2h\), \(a + 3h\), ... \(a + (n - 1)h\), \(b\). Derive the trapezoidal rule formula for \(n\) equal sub-intervals. Move the slider to the right to check your answer.
- What observation can you make regarding the approximate area as the number of sub-intervals \(n\) is increased?