Cylindrical Coordinates

Calculus 3 · Axiom Academy

LESSON Cylindrical Coordinates Polar coordinates in the plane, stacked with a height z — and the factor of r that rides along in every triple integral. A point is pinned down by three numbers: r , its distance from the z-axis; , the angle swept around the z-axis; and z , its ordinary height. The first two are exactly polar coordinates in the xy-plane. 2. The Volume Element: Why the r ? Switching a triple integral into cylindrical coordinates is not just a relabeling — we must account for how a little box of volume is stretched. A polar cell with the same and covers more area the farther it sits from the axis, because its arc edge has length . its volume, with the height dz Cylindrical coordinates shine when a boundary is naturally , , or . Three regions cover most of what you will meet — and every one keeps the r in . Solid cylinder — radius R , height h Full turn , radius , height — and the r integrand delivers the familiar . Cone — radius grows with height The radial bound is a function of z, so r is integrated first (innermost); that taper is what makes it a cone. Annular region — between two radii Only the radial limits change: r runs from the inner radius R_1 out to the outer radius R_2 . You can place a point in , you know why the volume element carries a factor of r , and you can set up the triple integral for the go-to regions. Scroll up to revisit any step.

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