\(\int_{0}^{z} (\sin x - \cos x) \, dx\) where \( z = \frac{\pi}{4} \), we have: \(\int_{0}^{\frac{\pi}{4}} (\sin x - \cos x) \, dx = \left[-\cos x - \sin x\right]_{0}^{\frac{\pi}{4}}\) Evaluating at the bounds gives: \(-\sqrt{2} - (-1) = -\sqrt{2} + 1\) Thus, the value of the integral is: \(\int_{0}^{\frac{\pi}{4}} (\sin x - \cos x) \, dx = 1 - \sqrt{2}\)