I know the value of the following definite integral

I also have a realtion

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

i.e. 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?