Original
We use different variables in that case, since we want {{MATH_247}} to be a function of time, {{MATH_248}}. With this adjustment, we define the accumulation function {{MATH_249}} as follows.
{{MATH_250}}
Since the integral
{{MATH_251}}
gives the displacement of the object on the time interval {{MATH_252}}, it follows that
{{MATH_253}}
where {{MATH_254}} gives the position of the object at the time {{MATH_255}}.
If we differentiate this equation with respect to {{MATH_256}}, we get that
{{MATH_257}}
Since {{MATH_258}}, we have that
{{MATH_259}}
This is the the First Fundamental Theorem of Calculus!