How do you find the area of a sphere in spherical coordinates?

How do you find the area of a sphere in spherical coordinates?

i.e. x^2+y^2 = R^2 with z between -R and R. Any region on the sphere has the same area as the corresponding area on the cylinder. The correspondence is via a radial projection out from the z axis. So, for example, the area between latitudes would be 2pi*R^2(cos(phi1)-cos(phi2)).

How do you write a sphere in cylindrical coordinates?

1 Answer

1. x2+y2+z2=R2 .
2. Since x2+y2=r2 in cylindrical coordinates, an equation of the same sphere in cylindrical coordinates can be written as.
3. r2+z2=R2 .

How do you find spherical and cylindrical coordinates?

To convert a point from spherical coordinates to cylindrical coordinates, use equations r=ρsinφ,θ=θ, and z=ρcosφ. To convert a point from cylindrical coordinates to spherical coordinates, use equations ρ=√r2+z2,θ=θ, and φ=arccos(z√r2+z2).

How do you find spherical coordinates?

To convert a point from cylindrical coordinates to spherical coordinates, use equations ρ=√r2+z2,θ=θ, and φ=arccos(z√r2+z2).

What is theta and rho?

Rho is the distance from the origin to the point. Theta is the same as the angle used in polar coordinates. Phi is the angle between the z-axis and the line connecting the origin and the point. The point (5,0,0) in Cartesian coordinates has spherical coordinates of (5,0,1.57).

Is RHO the radius of a sphere?

verifying that ρ= constant is the sphere of radius ρ centered at the origin.

How to find the area of a sphere using calculus?

With a few calculus, you can find d s = R R 2 − r 2 d r → d A = 2 π r R R 2 − r 2 d r This is a method to find the area of the sphere. Of course, a simpler method consists in doing the job in spherical coordinates instead of Cartesian coordinates.

How do you find the differential area in spherical coordinates?

To stay in spherical coordinates, you need to write the differential element of area in spherical coordinates. In the ϕ ^ direction, the differential arc is r d ϕ. In the θ ^ direction, the differential arc is r sin θ d θ, as you can convince yourself by drawing a diagram or looking in a calculus book. Thus the differential area is r 2 sin θ d θ.

Why can’t I Divide a sphere into disks?

There are two problems here. First: Not only do your disks change radius from the apex to the base of the hemisphere, but they do so at a varying rate. Let’s say (for concreteness) that the sphere is x 2 + y 2 + z 2 = R 2, and we’re dividing the sphere into disks based on the x -coordinate.

What is the area between latitudes on a sphere and cylinder?

Any region on the sphere has the same area as the corresponding area on the cylinder. The correspondence is via a radial projection out from the z axis. So, for example, the area between latitudes would be 2pi*R^2 (cos (phi1)-cos (phi2)). Mar 8, 2013