Line Integrals of Scalar Functions

Calculus 3 · Axiom Academy

LESSON Line Integrals of Scalar Functions Add up a function along a curve, not an interval — and find the mass of a bent wire whose density changes from point to point. A wire runs along a curve C . At each point it has a density f(x,y,z) (mass per unit length). To get the total mass, chop the wire into tiny pieces of arc length , weigh each piece as , and add them up. As the pieces shrink, the sum becomes an integral. mass = sum of (density) × (arc length) over the whole wire Geometrically this is the area of a curtain : stand a fence on the curve C whose height at each point is f . Straighten the wire out along its arc length s and the fence becomes an ordinary region — its area is exactly . The result is one number (total mass, total charge, or that curtain's area). To compute it, parametrize the curve as for . A uniform step dt in the parameter covers a piece of arc length — longer where the curve moves fast , shorter where it crawls. the speed factor ‖r′(t)‖ converts dt into arc length ds Do not drop the speed factor. is what turns the line integral into an ordinary single-variable integral in t : 4. Worked Example: A Helix Wire Find the mass of a wire shaped as the helix for , with density f(x,y,z)=z . Set up the speed factor first: Now evaluate the integral — with constant, the speed factor pulls straight out: You can now integrate a scalar function along a curve — weighting each piece by arc length, never forgetting the speed factor. Scroll up to revisit any step.

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