Q. 2 - Calculus

Question

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A region that, when revolved around the x-axis, has both disk and washer cross sections.

(b) A region that, when revolved around the y-axis, has both disk and washer cross sections.

(c) A solid of revolution for which it is not possible to use
the disk or washer method.

Step-by-Step Solution

Answer

(a) A region that, when revolved around the x-axis, has both disk and washer cross sections is $y=\frac{1}{2}\sqrt{x}$.

(b) A region that, when revolved around the y-axis, has both disk and washer cross sections is $y=\frac{1}{2}{x}^2$.

(c) A solid of revolution for which it is not possible to use
the disk or washer method is $y=(x-1){(x-2)}^2$.

Part (a) Step 1. Given Information.

A region that, when revolved around the x-axis, has both disk and washer cross sections

Part (a) Step 2. Give an example.

Consider the region $y=\frac{1}{2}\sqrt{x}$.

If it is revolved around x-axis, then it will disk shape.

Similarly consider the region between $y=\frac{1}{2}\sqrt{x} and y=\sqrt{x}$

When it is revolved around x-axis, then it will given washer shape.

Part (b) Step 1. Give an example.

Consider the region of the parabola, $y=\frac{1}{2}{x}^2$

If it is revolved around y-axis, then it will disk shape.

Similarly consider the region between $y=\frac{1}{2}{x}^2 and y=x^2$

When it is revolved around x-axis, then it will given washer shape.

Part (c) Step 1. Give an example.

From the given information, the equation of the curve should be of the form $x=f\left(y\right)$ .

Consider the equation where it is not possible or may not be easy.

$y=(x-1){(x-2)}^2$

in which it may not be possible to write equation in the form of y and the volume cannot be find using disk or washer methood.