Volume of a Sphere using Rectangular, Cylindrical, and Spherical Coordinates

Example: Find the volume, V, of a sphere of radius, R, first using rectangular coordinates.

Use a Type 1 region where the projection is on the x y - plane and integrate in the z-direction

first. See the figure below.

The projection of the sphere on to the xy-plane is a circle of radius R. Use symmetry

by evaluating 1/8^th of the volume and multiplying by 8 to get the total volume. Thus

the integral becomes

x = R y = √(R² – x²) z = √(R² – x² – y²)

V = 8 ∫ ∫ ∫ [ dz ] dy dx

x = 0 y = 0 z = 0

The first integration gives

x = R y = √(R² – x²)

V = 8 ∫ ∫ √(R² – x² – y²) dy dx

x = 0 y = 0

Notice that the integral on the variable y has the form ( a² – y²) where a² = R² – x².

So you could proceed by using a substitution y = a sin α . But at this point you might

instead switch to “polar coordinates” for the double integral. Then the integral becomes

θ = π/2 r = R

V = ∫ ∫ √(R² – r²) r dr dθ

θ = 0 r = 0

Let w = R² – r² so dw = - 2r dr and r dr = - ½ dw

Click here to continue with this example.

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