Limits for conditional variance of $X$ given $Y$?

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$$f(x,y)= 8xy \space\space\space\space\space\space\text { for } 0 \lt x \lt y \lt 1$$

What will be limits for integration in the formula for conditional variance of $X$ given $Y$?

Will it be from $x$ to $1$ or from $0$ to $y$?

Edit: I am getting the answer 0 upon inserting the limits 0 to y. Could it be correct?