Q 8 - Calculus

Question

In the reading we mentioned that the trapezoid sum is the average of the left sum and the right sum. Use the solutions of Examples 1 and 4 to show that for f (x) = x² − 2x + 2, [a, b] = [1, 3], and n = 4, the trapezoid sum is indeed the average of the left sum and the right sum.

Step-by-Step Solution

Answer

The average of the left and right sums is the trapezoid sum.

Step 1: Given function:

$f\left(x\right)=x^2-2x+2$

Step 2: Solution Explanation

The solutions to $f\left(x\right)=x^2-2x+2$are as follows:

Using the appropriate total, 

the area is $5.75$ square units when utilizing the right sum;

The area is $3.75$square units when utilizing the left sum;

$4.625$ square units are under the mid-point sum.

The trapezoid total area is $4.75$ square units.

Calculate the average of the left and right sums.

$$\begin{gathered} =\frac{5.75+3.75}{2} \\ =\frac{9.5}{2} \\ =4.75 \end{gathered}$$

As a result, the trapezoid sum equals the average of the left and right sums.

Hence Proved.