Integral Bounds and Sums

An investigation may supply rectangles and ask you to compare their total area with an integral. Read the rectangle heights and intervals from the diagram, then justify whether they lie above or below the curve.

For the increasing curve \(y=x^2\) on \(0\leq x\leq n\), a unit-width rectangle with right endpoint \(r\) has height \(r^2\). Each rectangle lies above the curve apart from its right corner.

\(\int_0^n x^2\,\mathrm{d}x<\sum_{r=1}^n r^2\), hence \(\dfrac{n^3}{3}<\sum_{r=1}^n r^2\) for positive integers \(n\).

For an upper bound on the sum, compare each height \(r^2\) with the curve over \([r,r+1]\):

\(\sum_{r=1}^n r^2<\int_1^{n+1}x^2\,\mathrm{d}x=\dfrac{n^3}{3}+n^2+n\).

Explain the inequality using the geometry before evaluating the integral. Follow any definitions supplied in the question; a memorised summation formula does not replace the requested argument.