Q. 38. - Calculus

Question

In Exercises 37–42, sketch the surface of revolution formed

when the given function on the specified interval is revolved

around the z-axis and find a function of two variables with the

surface as its graph.

$f\left(x\right)=\sqrt{x},[0,4]$

Step-by-Step Solution

Answer

The required surface formed is as shown below:

The function of two variables to represent the surface of revolution is determined by replacing  $\text{'}x\text{'}$ by $\sqrt{x^2+y^2}$is ${(x^2+y^2)}^{\frac{1}{4}}$

Step 1: Given information

The function is $f\left(x\right)=\sqrt{x}$over an interval of $[0,4]$

The $z$-axis is the center of this function. 

Step 2: The objective is to sketch the surface of the revolution

The function represents a parabola that passes through the $xy$-origin plane and along the $x$-axis.

When the $z$-axis of this parabolic form is rotated,

The surface formed is as shown below: 

Step 3: The objective is to find a function of two variables to represent this surface.

By replacing by the function of two variables to represent the surface of revolution is determined $\text{'}x\text{'}$by $\sqrt{x^2+y^2}$

$$\begin{gathered} f(x,y)=\sqrt{\left(\sqrt{x^2+y^2}\right)} \\ ={(x^2+y^2)}^{\frac{1}{4}} \end{gathered}$$