Advice and help on cable problem please

Hi,
I'm doing some self study on the subject of cables and I am curious about the reversed process of working out the linear mass density when given the shape of the cable (instead of the usual case of determining the shape of the cable given the density). The text I am currently reading explains an example but I would like to know how it works for other examples. If anyone can offer some help or advice, it would be brilliant!

Please find problem and my attempted solution below:



Many many thanks in advance!