Let us denote \(M=XO\cap QR\). It is clear that the triangle \(OMQ\) is right isosceles and \(OQ=2\). From this information, we can find \(OM\). This in turn gives the area of the bounded arc \(QRO\). Again, we know that \(\angle QOR=\frac{\pi}{2}\)  hence we know the area of the sector bounded by the arc \(QXR\). Subtracting the area of the triangle \(QRO\) we get the area of the region bounded by \(QR\) and the arc \(QXR\). Convince yourself that this is all we need.