Graphing x^2+y^2+z^2=z: Integral & Range Help

[kthouz]

can someone help me in graphing the following function: x^2+y^2+z^2=z.
i actually know about x^2+y^2+z^2=k (k=constant) but there above the radius z still changing. the qustion was to intagrate that function and i want to use the spherical coordinates then i will need to know the range of the radius.

 

[AiRAVATA]

Why don't you complete squares?

Hint: $(z-z/2)^2=z^2-z+1/4$

 

[tacman]

Sure, I can help you with graphing and integrating the function x^2+y^2+z^2=z. This function is known as a "sphere" or "spherical surface" and can be graphed in three-dimensional space.

To graph this function, we can use a graphing calculator or plot points manually. To plot points manually, we can choose different values for x and y, and then solve for z using the equation x^2+y^2+z^2=z. This will give us a set of points that lie on the surface of the sphere. We can then connect these points to get a continuous curve, which will represent the graph of the function.

To integrate this function, we can use spherical coordinates. In spherical coordinates, the radius is denoted by ρ, the polar angle is denoted by θ, and the azimuthal angle is denoted by φ. The range of the radius, ρ, will depend on the bounds of the integral. If the integral is over the entire sphere, then the range of ρ will be from 0 to the radius of the sphere. However, if the integral is over a smaller portion of the sphere, then the range of ρ will be from 0 to the radius of that portion.

To find the range of the radius for a specific integral, we can use the equation x^2+y^2+z^2=z and solve for ρ. This will give us the range of ρ in terms of the other variables. We can also use the Pythagorean theorem to find the radius of the sphere, which is √z.

I hope this helps you in graphing and integrating the function x^2+y^2+z^2=z. Let me know if you need any further clarification or assistance.