Surface Area

Calculus 3 · Axiom Academy

LESSON Surface Area in Multivariable Calculus How do we measure the area of a curved surface? See the geometric idea behind the surface-area formula, in motion. 1. The Problem: Flat vs. Curved Project a small rectangle from the xy -plane straight up onto the surface z=f(x,y) and it lands as a tilted patch . The steeper the surface, the more that patch is stretched relative to its flat shadow. Our whole job is to find that stretching factor. 2. Tangent Plane Approximation Over a tiny patch, the surface is almost flat — we can replace it with its tangent plane . Slide a point along the surface and watch the tangent tilt: its slope in the x -direction is , and in the y -direction it is . The patch is the parallelogram spanned by those two tangent vectors, so its area is the magnitude of their cross product . The three squared shadows of the patch — onto the xy -, xz -, and yz -planes — add up (Pythagoras) to dS^2 : 4. Example: Paraboloid z = x^2 + y^2 Find the surface area of the paraboloid z = x^2 + y^2 above the unit disk . Watch the surface area accumulate as we sweep outward in the radius r . f(x,y) = x^2 + y^2 , so f_x = 2x and f_y = 2y . In polar coordinates , this becomes . Integrate over the disk (remembering the polar area element ): Does the formula reproduce the known hemisphere area ? Sweep the disk of radius R outward and watch the accumulated surface area climb to exactly that value. The stretching factor simplifies neatly:

This is the written version of the interactive lesson above. See the full Calculus 3 course.