This article will derive the area of a circle and volume of a sphere using integration. It builds on a previous article introducing calculus, and aims to function as a primer on polar and spherical coordinate systems.
A circle is a 2D object with extent in both x and y. Its area is, of course, A = 𝜋r². It is very telling that this well-known equation relies not on x or y but on the radius, r. This is purely for simplicity. The radius is that number which intuitively defines a circle, it is the same all the way around, while x and y change drastically. To see how r relates to x and y consider a circle of radius r centred on the origin. Any point along the circumference, like the blue point below, can be treated as making a right-angle triangles with both axes, as shown below.
The sides are given by the trigonometrical relations (from r and the angle θ swept from the x-axis). In fact, any point can be specified using these coordinates r and θ; for example (2, 𝜋/6) refers to a point 2 units from the origin at an angle 𝜋/6 (or 30º) much like the one pictured above. θ ranges from 0 to 2𝜋 (or 360º) since it would simply repeat after one revolution anyway, while r can be any number greater than 0. This is the polar coordinate system.Finding the area of a circle in the x-y coordinate system requires integrating with respect to x and then with respect to y. Polar coordinates do the same thing but with r and θ. Double integrals may look intimidating at first sight but they are simply performed one at a time from the inside out.To solve in x and y is long and tedious, which is why area is commonly given in terms of r. A change to polar coordinates simplifies things, but at a cost. A conversion factor is needed, called the Jacobian. Fortunately, the change to polar requires a Jacobian of simply r; derived here. A more complex derivation follows later, so a glance at this one is advised if only to avoid the sensation that it’s being pulled from thin air. The area of a circle can thus be described as:Where θ ranges from 0 to 2𝜋, the angle swept by a circle, and r is limited between 0 and r. This is really the same as saying r isn’t defined, other than its being greater than 0. The integral of dθ = θ, so:θ then assumes the values of its limits, top minus bottom, so θ = 2𝜋 – 0, where the zero can be ignored:Then integrate r. Since it is undefined, i.e. r = r, its limits can be ignored.Finding the volume of a sphere requires a switch to the spherical coordinate system. Whereas polar coordinates specify a position on a circle, in spherical coordinates points are specified by their position on a sphere, using r, θ and φ. r is exactly as before, it points out from the origin and gives how far away the point is, or the radius of the sphere. But now two angles are needed, θ and φ, to specify a position. These are equivalent to latitude and longitude on the Earth: the first tells you how far South of the North pole you are, which of course isn’t enough to know your position, while the second tells you how far East you are from the Greenwich meridian. Latitude, θ, goes from 0 to 𝜋 (180º, i.e. North pole to South pole) while longitude, φ, goes from 0 to 2𝜋 (360º, i.e. Greenwich to Greenwich). These are shown below.
A triple integral is needed to derive the volume of a sphere, and has the form:In spherical coordinates these become drdθdφ and the Jacobian is r² sinθ, derived here. This gives the volume of a sphere as:Integration is much as before, and can in fact be performed in any order.Thus the volume of a sphere is given as expected: