I have a question that annoys my basic understanding of kinetic energy.

I know if I have a force-time plot then the area under the curve is equivalent to the impulse imparted on an object (in units Newton-Seconds). I know that this is also equivalent to the change in momentum of the object i.e. Ft = delta mV

I know that i can get these values from the plot by simply intgrating the Force function wrt to time. Now I want to determine the kinetic energy involved in this event. I know that k.e. = 1/2mV^2 so I know that is simply the integral of the impulse w.r.t. to

**velocity**
Now that is the part conceptually I don't really grasp. What does it really mean to integrate with respect to velocity? This means I am suddenly on a velocity

domain, not time, and I don't really understand this?

Thanks.