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A landscape gardener lays a curved border of stones along the graph of y = x²/4 - (ln x)/2, from x…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A landscape gardener lays a curved border of stones along the graph of $y = \dfrac{x^2}{4} - \dfrac{\ln x}{2}$ , from $x = 1$ to $x = 4$ , where $x$ and $y$ are in metres. The stones cost $12$ pounds for each metre of border. Find the length of the border and its cost.

Need a reminder? The key ideaShow

The length of the curve $y = f(x)$ from $x = a$ to $x = b$ is $L = \displaystyle\int_a^b \sqrt{1 + [f'(x)]^2}\,dx$ . It comes from adding the tiny hypotenuses $ds = \sqrt{dx^2 + dy^2}$ . Look for a perfect square under the root.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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