Q. 39. - 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.

$f\left(x\right)=x^2 ,\left[0,2\right]$

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 $(x^2+y^2)$

Step 1: Given information

The function is $f\left(x\right)=x^2$over an interval of $[0,2]$

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

$$\begin{gathered} f(x,y)=({\sqrt{x^2+y^2)}}^2 \\ =(x^2+y^2) \end{gathered}$$