Q. 18 - Calculus

Question

Suppose you wish to use n frustums to approximate the area of the surface obtained by revolving the graph of a function y = f (x) around the x-axis on [a, b]. Use a labeled graph to explain why the average radius r k of the kth frustum is given by,

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

Step-by-Step Solution

Answer

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

Step 1. Given information .

Consider the given statement . Suppose you wish to use n frustums to approximate the area of the surface obtained by revolving the graph of a

function y = f (x) around the x-axis on [a, b].

Step 2. Classify the average radius of frustum .

Since f is continuous and the average lies between $f\left(x_{k-1}\right)$ and $f\left(x_k\right)$, the Intermediate Value Theorem guarantees that there

is some point $x_k$ at which $f\left(x_k\right)$ is equal to $\frac{f\left(x_{k-1}\right)+f\left(x_k\right)}{2}$ .

Step 3. Notation of the frustum .