Q.18 - Calculus

Question

Use a labelled graph to demonstrate that the average radius $r_{k}ofthe{k}^{\text{th }}$frustum is given by if n frustums are used to approximate the area of the surface obtained by rotating the graph of a function $y=f\left(x\right)$ around the $x$-axis on $[a, b]$

$r_k=\frac{f\left(x_{k-1}\right)+f\left(x_k\right)}{2}$

Step-by-Step Solution

Answer

The function $y = f\left(x\right)$ on $[a,b]$ by using the graph average radius of$kth$frustum is $r_k=\frac{f\left(x_{k-1}\right)+f\left(x_k\right)}{2}.$

Step 1 : Given information

The function  $y = f\left(x\right)$ on $[a,b]$ the average radius$r_k \mathrm{of} \mathrm{the} k^{\text{th }}$frustum is given by $r_k=\frac{f\left(x_{k-1}\right)+f\left(x_k\right)}{2}$.

Step 2 : Calculation

The graph below displays the solid's $kth$frustum, which was created by rotating the function's graph around the $x$-axis.

The radii $A, C$, and $B$in the aforementioned graph correspond to the frustum's radii at$x_{k-1}and{x}_k$, respectively. Therefore, the average of the lengths $A, C,$ and $BD$ equals the average radius of the $kt h$ frustum.

As a result, the Average Radius of the $kth$ Frustum is given by,

$r_k=\frac{f\left(x_{k-1}\right)+f\left(x_k\right)}{2}.$