First graph the region of integration. Since x is between 0 and 1, draw vertical lines at x= 0 and x= 1. Since y is between x and , you want to graph those. Of course, y= x is the straight line from (0,0) to (1,1).
It's that is the problem? Square both sides to get so . Now complete the square in x by adding 1 to both sides: but so this is . The graph of that is circle with center at (1, 0) and radius 1. Because y was equal to the positive square root, we have the upper half circle and it should be easy to see that that intersects y= x at (0,0) and (0,1) and is above it.
Reversing the integral, y ranges from 0 to 1 and, for each y, from the left side of the graph on the left up to x= y on the right.
Solve for x. That is a quadratic and will give two solutions. Take the one that is less than 1 for y between 0 and 1.