91Ó°ÊÓ

The interval $[a,b]$on region $f\left(x\right)$

c is the midpoint of the interval $[a,b]$.

The points are localid="1660922247766" $A(a,0);B(b,0);C(c,f(c\left)\right);D(b,f(b\left)\right);E(a,f(a\left)\right);R(b,f(c\left)\right);S(a,f(c\left)\right)$

To show where these points are, draw a graph with the specified points on it.

Draw the trapezoid ABDE and shade its area by connecting the points.

The area under a curve $f\left(x\right),$ above the x-axis, on the interval $[a,b]$is said to be determined by integral ${∫}_a^{b}f\left(x\right)dx$.

To approximate this integral using the trapezoidal rule, the region under the function $f\left(x\right)$ can be divided into n-trapezoids.

Using this theorem, the above trapezoid can be said to represent the single trapezoidal approximation of the region bounded by $f\left(x\right)$in the interval $[a,b].$

The points are $R(b,f(c\left)\right)$$,S(a,f(c\left)\right)$

Mark the points $R(b,f(c\left)\right)$and $S(a,f(c\left)\right)$and name the area that the rectangle represents.

The region is divided into n-rectangles with heights equal to the value of the function at the midpoint of each interval in order to approximate an integral using the midpoint sum algorithm.

According to this definition, the rectangle shown above is a single rectangle used to approximate the midway of the integral ${∫}_a^{b}f\left(x\right)dx$.

The chord of the function $f\left(x\right)$in the interval $[a,b]$is represented by the line connecting D and E. The midpoint of the interval $[a,b]$is point C.

The midpoint theorem states that the tangent to a curve will be a line parallel to a chord that passes through the middle.

As a result, the curve at C will be tangent to the line through C that is parallel to D and E.

Create the trapezoid ABPQ by marking the points P with the x-coordinate ''b'' and Q with the x-coordinate ''a'' on this tangent line.

As a result, the area under the tangent at the interval's midpoint is represented by the trapezoid ABPQ.

The name "midpoint trapezoid" refers to the shape.

As a result, the area indicates the midpoint trapezoid method's approximation of the integral.

The line PQ

$A1$should be placed between the PQ line and the f graph.

This depicts the region that lies beneath the tangent line but not the graph. As a result, it would indicate the mistake from the midpoint tangent method's approximation.

The region would indicate the inaccuracy of the midpoint technique of approximation because the midpoint tangent method is similar to the midpoint method of approximation.

$A2$should be placed between the DE line and the f graph.

The area is covered by graph f but not by chord DE. As a result, it would indicate the approximation error from the trapezoid approach.

The chord line and the error indicated by the midpoint tangent method are the extreme error values, as was previously stated. The midpoint tangent line is the chord's limiting value.

Therefore, the trapezoid method error must fall within this range.

This clarifies the claim made in the theorem that gives the bounding values for the approximation errors produced by the trapezoidal method.