Q. 53 - Calculus

Question

Use Definition 4.8 to prove that if a function f is positive and concave up on $[a, b]$, then the trapezoid sum with n trapezoids is an always an over-approximation for the actual area.

Step-by-Step Solution

Answer

It is proved that if a function f is positive and concave up on $[a,b]$, then the trapezoid sum with n trapezoids is an always an over-approximation for the actual area.

Step 1. Given Information

We are given that if a function f is positive and concave up on $[a,b]$, then the trapezoid sum with n trapezoids is an always an over-approximation for the actual area.

Step 2. Proving the statement

The right sum defined for n rectangles on $[a, b]$ is $\sum _{k=1}^{n}f\left(x_k\right)\Delta x$.

Where, $\Delta x=\frac{b-a}{n},x_k=a+k\Delta x$.

The average of left sum and right sum is,

$\frac{{\sum }_{k=1}^{n}f\left(x_{k-1}\right)\Delta x+{\sum }_{k=1}^{n}f\left(x_k\right)\Delta x}{2}=\sum _{k=1}^n\frac{f\left(x_{k-1}\right)+f\left(x_k\right)}{2}\Delta x$

The function is concave up and is positive so the average of the left sum and the right sum will be over-approximation.

The trapezoid sum for n rectangles on $[a, b]$ is $\sum _{k=1}^n\frac{f\left(x_{k-1}\right)+f\left(x_k\right)}{2}\Delta x$.

Hence, the trapezoid sum approximation will also be an over approximation.