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The left sum is over-approximation when the function is decreasing monotonically on the interval$[a,b]$.

The trapezoid sum is over-approximation when the function is concave upwards on the interval$[a,b]$.

Thus, to have a function whose left sum and trapezoid sum is an over approximation, create a function which is concave upwards and decreasing on the interval .

Assume the graph of the function $y=x^2(x+2)$on the interval$[-4,3]$.

The graph is shown below:

The right sum is over-approximation when the function is increasing monotonically on the interval $[a,b]$.

The trapezoid sum is over-approximation when the function is concave upwards on the interval $[a,b]$.

Thus, to have a function whose right sum and trapezoid sum is an over approximation, create a function which is concave upwards and increasing on the interval $[a,b]$.

Assume the graph of the function $y=x^2(x+2)$on the interval $[-4,3]$.

The graph is shown below: