Find Change in output from marginal products of labor/capital
I think your solution is golden.

Taking the integral of the marginal products could only work if the marginal products were constant everywhere. This is not mentioned in the text, and in fact if you assume it to be constant then production given this level of $K$ and $L$ would not be 231.

Another interesting tidbit:
 If you knew the marginal product function of $L$ and also that of $K$ taking the integrals and summing them up would only work if $MPL$ is constant in $K$ and vice-versa.