To evaluate the integral \[ \int_{-2}^{1} (x - 1)^{2} \, dx, \] we will follow a step-by-step approach. ### Step 1: Expand the Integrand First, we need to expand the expression \((x - 1)^{2}\): \[ (x - 1)^{2} = x^{2} - 2x + 1. \] ### Step 2: Set Up the Integral Now, we can rewrite the integral with the expanded form: \[ \int_{-2}^{1} (x^{2} - 2x + 1) \, dx. \] ### Step 3: Break Down the Integral We can separate the integral into three simpler integrals: \[ \int_{-2}^{1} x^{2} \, dx - 2 \int_{-2}^{1} x \, dx + \int_{-2}^{1} 1 \, dx. \] ### Step 4: Evaluate Each Integral 1. **Evaluate \(\int_{-2}^{1} x^{2} \, dx\)**: The antiderivative of \(x^{2}\) is \(\frac{x^{3}}{3}\). Thus, \[ \int_{-2}^{1} x^{2} \, dx = \left[ \frac{x^{3}}{3} \right]_{-2}^{1} = \left( \frac{1^{3}}{3} - \frac{(-2)^{3}}{3} \right) = \left( \frac{1}{3} - \frac{-8}{3} \right) = \frac{1}{3} + \frac{8}{3} = \frac{9}{3} = 3. \] 2. **Evaluate \(-2 \int_{-2}^{1} x \, dx\)**: The antiderivative of \(x\) is \(\frac{x^{2}}{2}\). Thus, \[ -2 \int_{-2}^{1} x \, dx = -2 \left[ \frac{x^{2}}{2} \right]_{-2}^{1} = -2 \left( \frac{1^{2}}{2} - \frac{(-2)^{2}}{2} \right) = -2 \left( \frac{1}{2} - \frac{4}{2} \right) = -2 \left( \frac{1}{2} - 2 \right) = -2 \left( \frac{1 - 4}{2} \right) = -2 \left( \frac{-3}{2} \right) = 3. \] 3. **Evaluate \(\int_{-2}^{1} 1 \, dx\)**: The integral of \(1\) over an interval is simply the length of the interval. Thus, \[ \int_{-2}^{1} 1 \, dx = [x]_{-2}^{1} = 1 - (-2) = 1 + 2 = 3. \] ### Step 5: Combine the Results Now, we can combine the results of the three integrals: \[ 3 - 3 + 3 = 3. \] ### Conclusion The value of the integral \[ \int_{-2}^{1} (x - 1)^{2} \, dx = 3. \] ### Final Answer The correct option is **A.** \(-\frac{10}{3}\) is incorrect, **B.** \(7\) is incorrect, **C.** \(9\) is incorrect, and **D.** \(11\) is incorrect. The correct answer is **3**. ### Revision Summary - Expand the integrand before integrating. - Break down the integral into simpler parts. - Evaluate each integral separately and combine the results. - Always check the limits of integration carefully.