what is this integral?

[kochibacha]

I dont find any definition about definite integral matched to this type where the start point and endpoint are not the same (LHS from 0 to D₀ , RHS from 0 to infinity), surprisingly it still gives mathematically correct result

 

[HallsofIvy]

The limits of integration are different because you are integrating with respect to different variables. What is the relationship between t and \(\displaystyle D_B\)? When \(\displaystyle D_B= 0\) what is t? When \(\displaystyle D_B= D_0\) what is t? Or the other way around. When t= 0 what is \(\displaystyle D_B\)? As t goes to infinity what happens to \(\displaystyle D_B\)?

 

[kochibacha]

im still confused. i tried to visualize the left hand side by geometric point of view " sum of tiny slice of g(D)=1 in direction of D". I just don't understand how they relate to each other

 

[HallsofIvy]

Can you not answer my questions? What is the relation between t and \(\displaystyle D_B\)?