# 1283: Integration - Approximate integration - Trapezium Rule for Difficult Integrals

Functions such as $\color{blue}{y = {e^{ - {x^2}}}}$ and $\color{blue}{y = \sin \left( {x^2} \right)}$
cannot be integrated by basic methods.

However, their definite integrals can be estimated using numerical methods such as the trapezium rule (or trapezoidal rule in the USA) which is presented here.

The method consists of dividing the area into strips. The area of each strip is estimated by taking it to be approximately a trapezium.

With the display, you should see how the accuracy of the estimate increases as the number of intervals increases.

With the display, you should also see how the trapezium rule overestimates where the curve is concave upwards, and underestimates where the curve is convex upwards.

However, their definite integrals can be estimated using numerical methods such as the trapezium rule (or trapezoidal rule in the USA) which is presented here.

The method consists of dividing the area into strips. The area of each strip is estimated by taking it to be approximately a trapezium.

With the display, you should see how the accuracy of the estimate increases as the number of intervals increases.

With the display, you should also see how the trapezium rule overestimates where the curve is concave upwards, and underestimates where the curve is convex upwards.

Select function: y =