### Product Rule & Integration by Parts

This white rectangle has width $u$ and height $v$, both of which vary as functions of time, $t$. We can write the rates of change as $\dot{u}$ and $\dot{v}$.
 The area of the white rectangle, $A=$ In a short time interval, $\delta t$, we can assume that $\dot{u}$ and $\dot{v}$ are roughly constant, so… The width increases by $\delta u =$ The height increases by $\delta v =$ The area increases by approximately the shaded area, so $\delta A \approx$ The rate of change of the area of the rectangle, $\dfrac{dA}{dt} \approx \dfrac{\delta A}{\delta t} \approx$