Relating ∫yx dx and ∫xy dy for a curve y=f(x)

[JulieK]

Is there a formal relation that links
$\int yxdx$ OR $\int_{a}^{b}yxdx$
with
$\int xydy$ OR $\int_{a}^{b}xydy$
where $y=f(x)$ over the interval $x\in\left[a,b\right]$.

 

[mathman]

Is there a formal relation that links
$\int yxdx$ OR $\int_{a}^{b}yxdx$
with
$\int xydy$ OR $\int_{a}^{b}xydy$
where $y=f(x)$ over the interval $x\in\left[a,b\right]$.

For the indefinite integral (assuming everything is nice):
$\int xydy =\int xyy'dx$

For the definite integral there is a a similar relationship, but be careful with the limits. For the dx integral, the limits refer to x, while for the dy integral, the limits refer to y.