Evaluating definite integral

I know the value of the following definite integral



[itex]\int_{a}^{b}ydx[/itex]



I also have a realtion



[itex]x=f(y)[/itex]



i.e. x is an explicit function of y but I do not have y as an explicit

function of x. The relation between x and y is generally non linear.



Now I want to get the following definite integral



[itex]\int_{a}^{b}\left[\int ydx\right]xdx[/itex]



i.e. [itex]\int ydx[/itex] multiplied by x evaluated over the interval [a,b].



Is there an analytic (not numeric) way to evaluate this integral using

for example mean value or similar averaging technique?