Q 37. - Calculus

Question

In Exercise, 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.

$f\left(x\right)=x,\left[0,3\right]$

Step-by-Step Solution

Answer

$f\left(x,y\right)=\sqrt{x^2+y^2}$

Step 1. Given information

Function is $f\left(x\right)=x$.

Step 2. Explanation

Function is $f\left(x\right)=x$

The function represents a line with slope 1 that passes through the origin of the xy plane.

This line generates a conical funnel shape when rotated around the z-axis.

The shape formed will be simply a section of this conical shape because the interval is just across the first quadrant of the xy-plane.

Replace x by $\sqrt{x^2+y^2}$ to get a function of two variables that denotes the surface of revolution

So,

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