$$ \begin{array}{|c||c|c|c|c|c|} \hline x & 0 & 0.5 & 1 &1.5 & 2 \\ \hline f(x) & 0 &6 & 12 & 22 & 42 \\ \hline \end{array} $$

The table above gives selected values for a continuous function \(f\). If \(f\) is increasing over the closed interval \([0,2]\), which of the following could be the value of \(\int\limits_0^2 f(x)  dx\) ?

Summary Submit

For an increasing function, the left Riemann sum is an underestimate of the area under the curve. The right Riemann sum is an overestimate of the area under the curve.

To find the left Riemann sum, use the left endpoints with interval \(\Delta x = 0.5\) :

$$ \text{Area} = \sum_{i=1}^4 f(c_i) \Delta x_i =0(0.5)+6(0.5)+12(0.5)+22(0.5)$$ $$ = 0.5(0+6+12+22) = 20 $$

The right Riemann sum is found using the right endpoints:

$$ = 0.5(6+12+22+42) = 41 $$

The two values are the underestimate and overestimate of the area under the curve:

$$ 20 \lt \int\limits_0^2 f(x)  dx \lt 41 $$

Only \(\boxed{35}\) meets these restrictions.