The limits of integration are incorrect. Theta should be from -pi/2 to pi/2 as the region of integration is bounded by x = sqrt(49-y^2) and x = 0 (i.e. the right hemicircle of radius 7).
Your integration is also incorrect.
Separating the integration with respect to theta from r, we obtain immediately theta evaluated from -pi/2 to pi/2 which is equal to pi. Evaluating the integration with respect to r, not forgetting the Jacobian factor r, we obtain after the substitution u = -r^2, change of limits (including reversal due to the introduced negative sign): (pi)(1/2)(e^u) | (u1 = -49, u2 = 0) which gives (pi/2)(1-e^-49).