2021 H2 Maths Paper 2, Question 2

Since $y = f(x)$ is increasing on $\lbrack 1,5\rbrack$, as shown in the given sketch, using the value at the left endpoint of each strip as the rectangle’s height gives a rectangle that lies entirely below the curve throughout that strip — the curve only gets higher moving right within the strip. So the total area of these rectangles is less than the area of $A$:

Using the value at the right endpoint of each strip instead gives a rectangle that lies entirely above the curve throughout that strip, since $f$ is increasing and the right endpoint gives the largest value of $f$ in the strip. So this total is greater than the area of $A$:

With $f(x) = \frac{1}{20}x^{2} + 1$ and $h = 0.8$, the strip boundaries are $x = 1,\ 1.8,\ 2.6,\ 3.4,\ 4.2,\ 5$.

Lower bound, using the left-endpoint sum from part (a):

Upper bound, using the right-endpoint sum from part (b):

For the area under $y = g(x)$ to be less than the left-endpoint sum $\sum_{n = 0}^{4}g(1 + nh)\, h$, that sum must overestimate the true area. This happens when $g$ is decreasing on $\lbrack 1,5\rbrack$: the left endpoint of each strip then gives the largest value of $g$ within that strip, so each rectangle lies entirely above the curve.