Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 20 / a / Solution

Write $d=b-a$ and $h=d/n$. The required Riemann sum isUsing $\sum k=n(n+1)/2$ and the given sum of squares,An equivalent expression separates its limit and finite-$n$ corrections:ConsequentlyThe function is continuous, so the Riemann sums converge to its definite integral, the desired area under the parabola. Both corrections are positive when $0<a<b$, agreeing with the geometric fact that these right-endpoint rectangles overestimate an increasing curve.