# A= dv/dt

by vijay_singh
Tags: dv or dt
 Sci Advisor HW Helper PF Gold P: 12,016 Now, WHY can we utilize at times the dv=adt formula, in particular, WHERE is it usable? Answer: When doing integration with the technique called substitution of variables (i.e the "inverse" of the chain rule): Given a=dv/dt, we have, trivially: $$\int_{t_{1}}^{t_{2}}adt=\int_{t_{1}}^{t_{2}}\frac{dv}{dt}dt$$ But the right-hand side can, by the theorem of substitution of variables, be reformulated, giving the identity: $$\int_{t_{1}}^{t_{2}}adt=\int_{v(t_{1})}^{v(t_{2})}dv=\int_{v_{1}}^{v_{2 }}dv$$ Now, by IGNORING that the limits of integration actually refers to the limits of DIFFERENT variables, we "may say" that the "integrands" are equal, i.e, adt=dv! Thus, adt=dv should, at this stage of your education, be regarded as notational garnish (or garbage, if you like!)