Question

If $f$is monotonically increasing on $[a,b]$, which of the given approximations is guaranteed to be an over-approximation for localid="1650791372188" ${\int }_a^{b}f\left(x\right)dx$? (Select all that apply.)

(a) left sum

(b) right sum

(c) trapezoid sum

(d) midpoint sum

Short answer

Right sum is guaranteed to be an over-approximation for${\int }_a^{b}f\left(x\right)dx$.

Step-by-step solution

Step 1. Given information

$f$is monotonically increasing on$[a,b]$.

Step 2. Consider a function 'f' which is monotonically increasing on interval a,b.

The integral ${\int }_a^{b}f\left(x\right)dx$gives the area bounded between the curve and $x$-axis in this interval. It is represented by the following figure.

Step 3. Left sum:

For a monotonic increasing function, the rectangles for left sum will always have top edge below the graph. Thus, area of the rectangle will be below the area of graph. Hence for a monotonic increasing function, left sum is known to be an under-approximation and not an over-approximation.

Right sum:

For a monotonic increasing function, the rectangles for right sum will always have top edge above the graph. Thus, area of the rectangle will be more than the area under the graph. Hence for a monotonic increasing function, right sum is guaranteed to be an over-approximation.

Step 4. Trapezoid sum:

The nature of approximation in case of trapezoid sum depends on the concavity of the graph. In case the graph is concave up, the trapezoid sum is over-approximation, for concave down , it is under-approximation. The monotonicity of increasing and the decreasing nature is not of any concern in this sum.

Hence, trapezoid sum cannot be guaranteed to be an over-approximation in case of an increasing function.

Midpoint sum:

The nature of approximation in case of midpoint sum depends on the concavity of the graph. In case the graph is concave up, the midpoint sum is over-approximation, for concave down , it is under-approximation. The monotonicity of increasing and the decreasing nature is not of any concern in this sum.

Hence, midpoint sum cannot be guaranteed to be an over-approximation in case of an increasing function.