When estimating an integral using cylindrical coordinates we divide up a solid into small pieces of dimension Δr by Δθ by Δz.
The shaded portion has volume [π(r+Δr)2-π r2]Δz Δθ/(2π)=rΔrΔθΔz+0.5(Δr)2ΔθΔz, which is approximately rΔrΔθΔz when Δr, Δθ, and Δz are small. This is a visual aid for why adding an extra r is necessary when integrating with cylindrical coordinates. However, the precise reason is that the absolute value of the Jacobian of the transformation x = r cosθ, y = r sinθ, z = z, is r. Therefore, when we integrate a function f(r,θ,z) over a solid T in cylindrical coodinates, we use
We can also change the order of integration if T is a basic solid (the boundary is a finite number of continuous surfaces--see Calculus One and Several Variables 8th Ed. Salas/Hille/Etgen pg 999.) Thus the following integrals are all equivalent:
To determine the limits of integration, we take the outermost variable and work inward. Sketching the graph of the solid T is extremely helpful.
Here are the basic step for integrating in the order dr dθ dz. Other orders are similar.
The integral will have the general form
Back to Describing Surfaces Using Different Coordinate Systems.