Building blocks

# Is the Serpentine Lake really 40 acres? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Area under a curve

Some of the ideas in the Serpentine example can be used to find the area between a curve and the $x$-axis. They can also be used to find the area between curves. The result for the area between a curve and the $x$-axis, called the trapezium rule, is illustrated below.

In this graph the points are spread out evenly, so that the $x$-coordinates increase by $h$ each time. By approximating the area under the graph using trapezia, we have $\text{Area between graph and x-axis}\approx \dfrac{h}{2}\Big(y_{0}+2(y_{1}+y_{2}+ y_{3}+y_{4}+y_{5})+y_{6}\Big)$

What would the formula be if we had $n+1$ evenly spaced points?

How could the formula be written if we knew we had the graph of a function, $f(x)$?