Can the trapezium rule help us find the area under $y = x\ln x$?

Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

The continuous function \(f\) maps the real interval \[\begin{equation*} a \le x \le b \end{equation*}\]

into a subset of the positive real numbers. Derive the trapezium rule for the approximate evaluation of the area under the curve \(y = f(x)\) between \(x=a\) and \(x=b\), using \((n-1)\) points of subdivision at a distance \(h = (b-a)/n\) apart.

Hence obtain an approximation to the integral \[\begin{equation*} I = \int_3^5 x \ln x \:dx \end{equation*}\]

using the points of subdivision given in the following table:

\(x\)	\(3.0\)	\(3.5\)	\(4.0\)	\(4.5\)	\(5.0\)
\(x \ln x\)	\(3.296\)	\(4.385\)	\(5.545\)	\(6.768\)	\(8.047\)