AP Calculus BC / Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
9.5 Integrating Vector-Valued Functions
Notes
Integrate each component separately, with a constant vector:
$$\vec r(t)=\vec r(0)+\int_0^t\vec v(s)\,ds$$
Practice
$\vec v(t)=\langle 2t,3\rangle$ and $\vec r(0)=\langle1,0\rangle$. $\vec r(2)=$
$\langle1+4,\,0+6\rangle$.
$\displaystyle \int\langle\cos t, 1\rangle\,dt=$
Componentwise antiderivatives.
$\displaystyle \int_0^1\langle 3t^2,2\rangle dt=$
$\langle t^3,2t\rangle$ from 0 to 1.
Ready for more? Take the Unit 9 test.