Concept preview

What you will be able to do

1. 01Integrate a vector-valued function componentwise, writing $\int\vec r(t)\,dt=\left\langle\int x(t)\,dt,\;\int y(t)\,dt\right\rangle+\vec C$ with a constant VECTOR of integration.
2. 02Recover a position function from a velocity function by integrating and using an initial condition $\vec r(t_0)=\vec r_0$ to solve for the constant vector.
3. 03Compute the net displacement over $[a,b]$ as the definite integral $\int_a^b\vec v(t)\,dt=\vec r(b)-\vec r(a)$, and distinguish it from indefinite integration.

A vector-valued function packages two ordinary functions into one moving arrow. Writing $\vec r(t)=\langle x(t),\,y(t)\rangle$ means that at each time $t$ the tip of the vector sits at the point $\left(x(t),\,y(t)\right)$, and as $t$ runs the tip traces a curve in the plane. Earlier in Unit 9 you differentiated such a function by differentiating each component: $\vec r\,'(t)=\langle x'(t),\,y'(t)\rangle$ gives the velocity vector. Integration simply runs that machine in reverse. Because the derivative acts component by component, so does the antiderivative — to integrate a vector-valued function you integrate each component separately and reassemble the results into a new vector.