Cylindrical and Spherical Surfaces

Calculus 3 · Axiom Academy

LESSON Cylindrical and Spherical Surfaces How simple 2D curves generate 3D surfaces — by translating them along an axis, or revolving them around one. 1. Cylindrical Surfaces: Translate a Curve A cylindrical surface is generated by taking a curve in one plane and translating it parallel to an axis. The key property: one coordinate is completely independent of the others — it never appears in the equation, so it is free to be anything. The generating curve — a circle of radius r , held fixed Every height z gets its own copy of that circle x^2 + y^2 = r^2 — circular cylinder parallel to the z -axis y = x^2 — parabolic cylinder parallel to the z -axis x^2 + z^2 = 4 — circular cylinder parallel to the y -axis 2. Surfaces of Revolution: Spin a Curve A surface of revolution is generated by rotating a curve around an axis. Every point on the curve traces out a circle as it turns, and its distance from the axis never changes. The sphere is a special case of this. The distance of each point from the origin. Every point of the semicircle sits at , and spinning never changes it. The rotation angle around the axis, running once all the way from 0 to . Slice perpendicular to the x -axis at x = c and you get the circle y^2 + z^2 = [f(c)]^2 — a circle of radius f(c) . Squaring kills the square root, and the leftover -x^2 moves across to give x^2 + y^2 + z^2 = r^2 .

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