I know the value of the following definite integral

$\displaystyle \int_{a}^{b}ydx$

I also have a realtion

$\displaystyle x=f(y)$

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

$\displaystyle \int_{a}^{b}\left[\int ydx\right]xdx$

i.e. $\displaystyle \int ydx$ 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?