Question

In Exercises 31-36, find an equation for the surface of revolution generated by revolving the curve in the indicated coordinate plane about the given axis. $$ 2 z=\sqrt{4-x^{2}} \quad x z \text { -plane } \quad x \text { -axis } $$

Short answer

The equation of the surface of revolution is \(\[x^{2} + y^{2} + z^{2} = 1\].\)

Step-by-step solution

Rewrite the equation

First, the equation needs to be rewritten. Divide both sides of the equation \(\[2z = \sqrt{4 - x^{2}}\]\) by 2 to get: \[z = \frac{1}{2} \sqrt{4 - x^{2}}.\] This equation resembles a semi-circle with a radius of 2/2=1 in the xz plane, centered at the origin along the x-axis.

Represent the revolving curve

Now, revolve this semi-circle about the x-axis. When revolving a two-dimensional curve in the xz plane around the x-axis, y-values change while x and z values stay the same. Given the symmetry of the semi-circle, a full-circle in three dimensions (a sphere) will be generated.

Formulate the equation of the sphere

For a sphere with a center at the origin (0,0,0) and radius r in a three-dimensional space, the equation is given as \[x^{2} + y^{2} + z^{2} = r^{2},\] The given curve revolves to form a sphere of radius 1; hence the equation of the surface of revolution becomes \[x^{2} + y^{2} + z^{2} = 1^{2}.\]