differential geometry - Surface Element in Spherical Coordinates ...
We are trying to integrate the area of a sphere with radius r in spherical coordinates. The angle θ runs from the North pole to South pole in radians. Angle θ equals zero at North pole and π at South pole. The distance on the surface of our sphere between North to South poles is rπ (half the circumference of a circle).
how to find surface area of a sphere - Mathematics Stack Exchange
and sum the area of all the shaded portions as t → 0 noting that δt = Rδθ. A = ∫2π 0 2RRdθ. or. A = 4πR2. Or more directly A = ∫2πRt = 02Rdt. † this is the same as the area of the 2R × t face of the wedge which leads to a simple proof that the surface area of a sphere is the same as that of its surrounding cylinder.
Surface Integral over a sphere - Mathematics Stack Exchange
But there are 2 typos. (1) x2 +y2 +z2− −−−−−−−−−√ = 2 x2 +y2 +z2 =4 x 2 + y 2 + z 2 = 2 x 2 + y 2 + z 2 = 4; (2)since S S is the sphere of radius 2 2 centred at the origin. (3) it is customary to use θ θ and ϕ ϕ for the polar and azimuth angles. The answer is correct and, actually, no integration is required: ∬S fdS ...
geometry - Surface area of a sphere with integration of disks ...
What you can do is use spherical coordinates, where each point in space is described by a radius r, and two angles θ, ϕ as in the link. In fact, to get the area of the sphere you need to keep the radius constant, and integrate over the angles that parametrize the given sphere. The result yields. S = ∫π 0∫2π 0 R2sinθdϕdθ = R22π∫π ...
Rate of Change of Volume in a Sphere - Mathematics Stack …
The volume of the sphere is: V = 4π 3 r3 V = 4 π 3 r 3. Differentiating volume with respect to radius gives: dV dr = 4πr2 d V d r = 4 π r 2. However, we want the differential of volume with respect to time. To do this we can multiply both sides by dr dt d r d t: dr dt dV dr = 4πr2dr dt d r d t d V d r = 4 π r 2 d r d t.
geometry - Area of surface between two lines of latitude?
Apr 15, 2021 · Start with a sphere of radius R, then take the part of the surface that exists between any two lines of latitude. I drew a pic with two examples, one near the equator and one near the pole. What is the area of this blue section of surface? I don't know what it's proper name is, maybe a band or zone.
geometry - Why do 4 circles cover the surface of a sphere ...
Is there a geometric explanation for why a sphere has surface area $4 \\pi r^2$ ? Ie equal to 4 times its cross-section (a circle of radius r).
geometry - Surface Area is the derivative of Volume, what is the ...
Aug 4, 2017 · Imagine a sphere, then imagine painting it. The amount of paint you use is based on the surface area, and becomes part of the volume of a new sphere. The rate of change of the volume of the sphere is equal to the surface area of the sphere. The outside of the paint is the new boundary of the sphere, and the inside of the paint is added to the ...
Why the surface of the sphere is not a Euclidean space?
The surface of a sphere is non-euclidean because the nearest analogs to straight lines, create circles, cross TWICE and no great circles can be parallel. Also, the circumference of any circle on the surface of a sphere is NOT 2 *pi * the radius, but is always SMALLER. Share.
calculus - Derive the surface of a sphere using integration ...
Mar 20, 2020 · The surface integral is, ∫S2π√r2 − z2ds. where the surface element ds is given by ds = rdθ, with θ being the angle between the radial direction and the z -axis. Note that cosθ = z r and. ds = rdθ = − ds sinθ = − rdz √r2 − z2. Plus into the area integral to …