Question

Suppose $f$is concave down on $[a,b]$. Use pictures to explain why the error incurred from using the trapezoid sum with $n$trapezoids to approximate ${\int }_a^{b}f\left(x\right)dx$will always be less than or equal to the difference of the trapezoid sum with $n$trapezoids and the midpoint sum with $n$rectangles.

Short answer

Draw the combined graph of the trapezoid sum and the midpoint sum as shown in the figure below:

From the graph, it is prove that the error from the Trapezoid sum is $\left|E_{TRAP\left(n\right)}\right|\le \left|TRAP\left(n\right)-MID\left(n\right)\right|$.

Step-by-step solution

Given information

The function $f$ is concave down on $[a,b]$.

Draw the graph of the area bounded between the function and x-axis on the interval [a, b].

The integral ${\int }_a^{b}f\left(x\right)dx$ denotes the area which is bounded between the curve and $x$-axis on $[a,b]$.

Asume the function $f\left(x\right)=\cos x+1.83$.

Draw the graph of the area bounded between the function and localid="1650027018016" $x$-axis as shown below:

Draw the five-rectangle trapezoid sum on [a, b].

Consider the number of intervals $n=5$.

For the concave-down function localid="1650026785087" $f\left(x\right)=\cos x+1.83$, the trapezoid sum must be less than the actual area under the curve, as shown in the figure below:

Draw the six-rectangle midpoint sum on [a, b].

Consider the number of intervals $n=5$.

For the concave-down function localid="1650026715733" $f\left(x\right)=\cos x+1.83$, the midpoint sum must be greater than the actual area under the curve as shown in the figure below:

Prove that the error incurred from using the trapezoid sum with n trapezoids to approximate ∫abf(x)dx will always be less than or equal to the difference of the trapezoid sum with n trapezoids and the midpoint sum with n rectangles.

The objective is to prove that $\left|E_{TRAP\left(n\right)}\right|\le \left|TRAP\left(n\right)-MID\left(n\right)\right|$.

Consider the number of intervals $n=5$.

The actual area area sandwiched between the smaller trapezoid sum and the larger midpoint sum as shown in the figure below:

From the graph observed that the actual area under the graph of a consistently concave-down function will always be between the trapezoid sum and the midpoint sum.

So, the error for Trapezoid of those sums must be less than the difference of the trapezoid sum and the midpoint sum, regardless the value of $n$.

This implies that $\left|E_{TRAP\left(n\right)}\right|\le \left|TRAP\left(n\right)-MID\left(n\right)\right|$.

Hence, it is proved.