A rigid rectangular loop of wire with width \(w\) along the \(x\)-axis and length \(L\) along the \(y\)-axis lies in the \(xy\)-plane and moves in the \(+x\)-direction at a constant speed \(v\). The loop moves through a non-uniform magnetic field directed perpendicular to the \(xy\)-plane, given by \(\vec{B}(x) = \beta x^2 \hat{k}\), where \(\beta\) is a positive constant. At an instant when the trailing edge of the loop is at position \(x\), which of the following pairs correctly identifies the integral setup for the magnetic flux \(\Phi_B\) through the loop and the magnitude of the instantaneous induced electromotive force \(\mathcal{E}\)?