Question

The error bounds for right and left sums in Theorem $5.25$apply only to monotonic functions. Suppose f is a positive integrable function that is increasing on $[a,c]$and decreasing on $[c,b]$, with $a<c<b$. How could you use a right or left sum to make an estimate of ${\int }_a^{b}f\left(x\right)dx$and still get a bound on the error? Draw a picture to help illustrate your answer.

Short answer

A picture which helps to illustrate my answer is,

Add the two error bounds of intervals, to determine the bound of error of complete interval.

$$\begin{gathered} \left|E_{LEFT\left(n\right)}^{\left[a,b\right]}\right|=\left|E_{LEFT\left(n\right)}^{\left[a,c\right]}\right|+\left|E_{LEFT\left(n\right)}^{\left[c,b\right]}\right| \\ =\left|f\left(c\right)-f\left(a\right)\right|\frac{\left(c-a\right)}{n}+\left|f\left(b\right)-f\left(c\right)\right|\frac{\left(b-c\right)}{n} \end{gathered}$$

Step-by-step solution

Step 1. Given information

Suppose $f$is a positive integrable function that is increasing on $[a,c]$and decreasing on $[c,b]$, with $a<c<b$.

Step 2. Sketch a curve representing the area under the graph of 'f' between the intervals a,c and c,b.

Step 3. Create the left sum rectangles to approximate the area under the graph of function 'f'.

Step 4. As seen in the above figure, the left sum is an under-approximation for the interval a,c,

while as it is an over-approximation for the interval $\left[c,b\right]$.The bound on the error for left or right sum approximation over the interval $\left[x_{k-1},x_k\right]$is given as $\left|E_{LEFT\left(n\right)}\right|=\left|f\left(x_k\right)-f\left(x_{k-1}\right)\right|∆x$.

Use this theorem to determine the bound on the errors of left sum approximation over the first interval of $\left[a,c\right]$.

role="math" localid="1650711139547" $$\begin{gathered} \left|E_{LEFT\left(n\right)}^{\left[a,c\right]}\right|=\left|f\left(c\right)-f\left(a\right)\right|\left(∆x\right) \\ =\left|f\left(c\right)-f\left(a\right)\right|\frac{\left(c-a\right)}{n} \end{gathered}$$

Step 5. Use this theorem to determine the bound on the errors of left sum approximation over the second interval of c,b.

$$\begin{gathered} \left|E_{LEFT\left(n\right)}^{\left[c,b\right]}\right|=\left|f\left(b\right)-f\left(c\right)\right|\left(∆x\right) \\ =\left|f\left(b\right)-f\left(c\right)\right|\frac{\left(b-c\right)}{n} \end{gathered}$$

The total area under the graph in the interval $\left[a,b\right]$can be said to be the sum of the two areas formed in the intervals $\left[a,c\right]$and $\left[c,b\right]$. Hence, the errors and their bounds can also be determined as the sum of the two errors defined above.

Add the two error bounds of intervals, to determine the bound of error of complete interval.

$$\begin{gathered} \left|E_{LEFT\left(n\right)}^{\left[a,b\right]}\right|=\left|E_{LEFT\left(n\right)}^{\left[a,c\right]}\right|+\left|E_{LEFT\left(n\right)}^{\left[c,b\right]}\right| \\ =\left|f\left(c\right)-f\left(a\right)\right|\frac{\left(c-a\right)}{n}+\left|f\left(b\right)-f\left(c\right)\right|\frac{\left(b-c\right)}{n} \end{gathered}$$