2015 H2 Maths Paper 1, Question 3

Approach: Interpret the sum as an area. Split $\lbrack 0,\ 1\rbrack$ into $n$ strips each of width $\frac{1}{n}$, and build a rectangle on each strip whose height is the value of $f$ at the strip’s right endpoint.

is the total area of $n$ rectangles, each of width $\frac{1}{n}$, with the $k$-th rectangle having height $f\left( \frac{k}{n} \right)$, as shown:

This sum is the total shaded area, which approximates the area under $y = f(x)$ from $x = 0$ to $x = 1$. As $n \rightarrow \infty$, the width $\frac{1}{n}$ of each strip $\rightarrow 0$, so the rectangles fit the curve more and more closely, and (since $f$ is continuous, so has no jumps for the rectangles to overshoot or undershoot indefinitely) the total rectangular area converges to the exact area under the curve.

Approach: Write the given limit in the standard form of part (i) by pulling $\sqrt[3]{n}$ inside each term of the sum.

This is exactly the sum in (i) with $f(x) = \sqrt[3]{x} = x^{\frac{1}{3}}$. Hence by (i),