Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 17 / a

For $b>0$, split the interval into $n$ equal widths $\Delta x=b/n$. Since $x^2$ increases on this interval, the left and right rectangles give lower and upper Riemann sums:Using the finite sum of squares,The rectangle bounds differ by $U_n-L_n=b^3/n\to 0$. The squeeze theorem therefore gives, directly from elementary areas,For $b=0$ the result is zero. If negative $b$ is allowed, reverse the integration limits and use the reflection $x=-u$; the same signed formula holds. No antiderivative is needed for this first-principles calculation.