Substitution Rule for Integrals: Solving for the Unknown Variable

[dilasluis]

Hello! My problem is the following:

Is

$\int_a^b f(z) dt = \int_{g(a)}^{g(b)} f(z) \frac{1}{g} dz$

?

$\frac{dz}{dt} = g$

Thank you!

 

[hunt_mat]

No. Is z a function of t?

 

[HallsofIvy]

If z is a an invertible function of t such that dz/dt= g(t), then dz= g(t)dt, dt= (1/g(t))dz, but you cannot have g(t) in the integral with respect to z.

 

[dilasluis]

z is a function of t, but not explicit, actually

$V_z = \frac{dz}{dt}$

was the relation from which we took $d t = \frac{dz}{V_z}$.

$V_z = cte$

 

[dilasluis]

My biggest problem with this question is $f(z)$ in both sides of the equation... and how do I change the integral from left side to the right.