91Ó°ÊÓ

Given,

The concave down in interval $[a,c]$ and concave up in $[c,b]$, with $a<c<b$

Assume a positive integrable function 'f' 'with $a<c<b$that is concave down in the interval $[a,c]$and concave up in the interval $[c,b]$.

The area bounded by the curve and the x-axis in this interval is given by the integral ${∫}_a^{b}f\left(x\right)dx$.

The goal is to find the upper bound on the trapezoid or midpoint sum error in order to estimate ${∫}_a^{b}f\left(x\right)dx$.

Draw a curve between$[a,c]$and $[c,b]$that represents the area under the graph of f '.

Using the trapezoidal rule, divide the region into trapezoids of fixed widths to approximate the area beneath the graph of function f .

The trapezoid sum is an under-approximation for the interval $[a,c]$, whereas it is an over-approximation for the interval localid="1661256313679" $[c,b]$, as seen in the diagram above.

Over the interval $\left[x_{k-1},x_k\right]$, the bound on the error for trapezoid sum approximation $\left|E_{\text{TRAP(n)}}\right|≤\frac{M{\left(x_k-x_{k-1}\right)}^3}{12n^2}$

Determine the bound on trapezoid sum approximation errors across the initial interval of localid="1661256322850" $[a,c]$using this theorem.

$\left|E_{\text{°Õей±ð)}}^{[a,e]}\right|≤\frac{M(c-a{)}^3}{12n^2}$

Determine the bound on trapezoid sum approximation errors across the second interval of localid="1661256298706" $[c,b]$using this theorem.

$\left|E_{\text{ткмץ)}}^{[c,b]}\right|≤\frac{M(b-c{)}^3}{12n^2}$

The sum of the two areas created in the intervals $\left[a,c\right]$and $[c,b]$can be considered to represent the total area under the graph in the interval $\left[a,b\right]$. As a result, the sum of the two mistakes mentioned above can be used to determine the errors and their boundaries.

To get the bound of error of a whole interval, add the two error bounds of intervals.

$\left|E_{TRAP\left(n\right)}^{\left[a,b\right]}\right|=\left|E_{TRAP\left(n\right)}^{\left[a,c\right]}\right|+\left|E_{TRAP\left(n\right)}^{\left[c,b\right]}\right|$

Likewise, the sum of two limits can be used to determine the error of approximation by the midpoint rule.

localid="1654844466792" $\left|E_{MID\left(n\right)}^{\left[a,b\right]}\right|=\left|E_{MID\left(n\right)}^{\left[a,c\right]}\right|+\left|E_{MID\left(n\right)}^{\left[c,b\right]}\right|$