To evaluate the integral \[ \int_{0}^{z} (\sin x - \cos x) \, dx \] where \( z = \frac{\pi}{4} \), we will follow these steps: ### Step 1: Find the Antiderivative First, we need to find the antiderivative of the integrand \( \sin x - \cos x \). 1. The antiderivative of \( \sin x \) is \( -\cos x \). 2. The antiderivative of \( -\cos x \) is \( -\sin x \). Thus, the antiderivative of \( \sin x - \cos x \) is: \[ -\cos x - \sin x \] ### Step 2: Evaluate the Definite Integral Now we will evaluate the definite integral from \( 0 \) to \( z \): \[ \int_{0}^{z} (\sin x - \cos x) \, dx = \left[-\cos x - \sin x\right]_{0}^{z} \] Substituting \( z = \frac{\pi}{4} \): \[ = \left[-\cos\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{4}\right)\right] - \left[-\cos(0) - \sin(0)\right] \] ### Step 3: Calculate the Values 1. **Evaluate at \( z = \frac{\pi}{4} \)**: - \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) - \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) Therefore: \[ -\cos\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = -\sqrt{2} \] 2. **Evaluate at \( x = 0 \)**: - \( \cos(0) = 1 \) - \( \sin(0) = 0 \) Therefore: \[ -\cos(0) - \sin(0) = -1 - 0 = -1 \] ### Step 4: Combine the Results Now we combine the results from the evaluations: \[ \int_{0}^{\frac{\pi}{4}} (\sin x - \cos x) \, dx = \left(-\sqrt{2}\right) - \left(-1\right) = -\sqrt{2} + 1 \] This simplifies to: \[ 1 - \sqrt{2} \] ### Conclusion: Final Answer Thus, the value of the integral is: \[ 1 - \sqrt{2} \] ### Correct Option The correct option is **D**: \( 1 - \sqrt{2} \). ### Explanation of Other Options - **Option A: \( \sqrt{2} + 1 \)**: This is incorrect because it does not match our derived result. The signs and terms do not align with the evaluation of the integral. - **Option B: \( \sqrt{2} - 1 \)**: This is also incorrect. While it has the correct components, the signs are wrong, leading to a different value. - **Option C: \( -\sqrt{2} - 1 \)**: This option is incorrect as it does not reflect the correct evaluation of the integral. The evaluation at \( z = 0 \) was miscalculated in this option. ### Revision Summary - The integral \( \int_{0}^{z} (\sin x - \cos x) \, dx \) was evaluated using the antiderivative. - The antiderivative of \( \sin x - \cos x \) is \( -\cos x - \sin x \). - Evaluating from \( 0 \) to \( \frac{\pi}{4} \) gives \( 1 - \sqrt{2} \). - The correct answer is \( 1 - \sqrt{2} \), corresponding to option D.