Review question

# 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

Ref: R7903

## Question

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:

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