Integration of Vector Functions

Calculus 3 · Axiom Academy

LESSON Integration of Vector Functions Integrate a vector function one component at a time — and watch the antiderivative rebuild position from velocity. Three ordinary integrals, packaged as one vector One arbitrary constant per component Each component is integrated independently, and we must include a vector constant of integration , where each C_i is an arbitrary constant. Nothing couples the components: the parameter t is shared, but the three integrals never talk to each other. 2. Example: Computing an Integral The result is a family of vector functions, parameterized by the vector constant . Slide C_1 and only the x -component moves; slide all three and you get a different member of the family. Every one of them has the same derivative, . In physics, we often know the velocity of an object and its initial position , and we want to find its position function . This is an initial value problem (IVP). Each little step of the path is — a scaled copy of the velocity vector. Integrating componentwise recovers the position — but only up to a constant vector. With determined, exactly one curve through space solves the problem. The same function, as a velocity Take — the vector function we just integrated — with the initial position . Since already equals at t = 0 , the initial condition gives , so . 4. The Constant Vector Is Where You Start

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