Can I reverse the PdV integral for work?

[Hercuflea]

Homework Statement

I have a problem where I know P1, P2, V1, and molecular mass. I do not know total mass, final volume, temperature 1 or temperature 2, or the work. I also know that P = 40V^2 + 20

Can I reverse the W = ∫PdV equation into the form W = ∫V dP? Because I could just solve the equation for V = sqrt(P/40 - 1/2) and integrate that. Can I do this?

Homework Equations

W = ∫PdV

The Attempt at a Solution

Qdot - Wdot = mdot ((u2 + v^2/2 + gz2) - (u1 + v^2/2 +gz1))
no heat transfer, constant mass, no change in altitude, no velocity. The system is being compressed.

 

[haruspex]

Can I reverse the W = ∫PdV equation into the form W = ∫V dP? .

Not quite. Integration by parts: $\int_a^by.dx + \int_{y(a)}^{y(b)}x.dy = [xy]_{x=a}^{x=b}$