Website title: Formal Relation Between Integrals of y=f(x) and y=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.