Q. 17 - 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 slant length s k of the kth frustum is given by ,

$s_k=\sqrt{{\left(\delta x\right)}^2+{\left(\delta y_k\right)}^2}$

Step-by-Step Solution

Answer

Slant height of the frustum $s_k=\sqrt{{\left(\delta x\right)}^2+{\left(\delta y\right)}^2}$ .

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. Find slant height .

The graph shown next at the left. Revolving this graph around

the x-axis produces the surface in the middle figure. One way to approximate the area of this surface is by approximating the graph with two straight line segments and then revolving these line segments around the x-axis to obtain truncated cones, as illustrated in the rightmost figure The truncated cones in this example are called frustums.

We must find the surface area of each frustum and then add up all those surface areas.Following the same algebra as we used for arc length, we can calculate the slant length of each frustum:

$s_k$ = slant length of kth frustum $=\sqrt{{\left(\delta x\right)}^2+{\left(\delta y\right)}^2}$

Step 3. Sketch the graph .

The graph for the function $y=f\left(x\right)$ is shown below .