To evaluate the integral \(\int^{2} _{3}(x^2 - 2x)dx\), we will follow a systematic approach. Let's break it down step-by-step. ### Step 1: Understand the Integral The integral \(\int^{2} _{3}(x^2 - 2x)dx\) represents the area under the curve of the function \(f(x) = x^2 - 2x\) from \(x = 3\) to \(x = 2\). However, notice that the limits of integration are from 3 to 2, which means we are integrating in the reverse direction. This will affect the sign of our result. ### Step 2: Find the Antiderivative To evaluate the integral, we first need to find the antiderivative of the function \(f(x) = x^2 - 2x\). 1. **Antiderivative Calculation**: - The antiderivative of \(x^2\) is \(\frac{x^3}{3}\). - The antiderivative of \(-2x\) is \(-x^2\). - Therefore, the antiderivative \(F(x)\) of \(f(x)\) is: \[ F(x) = \frac{x^3}{3} - x^2 + C \] where \(C\) is the constant of integration (not needed for definite integrals). ### Step 3: Evaluate the Definite Integral Now we will evaluate the definite integral using the Fundamental Theorem of Calculus, which states that: \[ \int_a^b f(x) \, dx = F(b) - F(a) \] In our case, we need to evaluate \(F(2)\) and \(F(3)\): 1. **Calculate \(F(2)\)**: \[ F(2) = \frac{2^3}{3} - 2^2 = \frac{8}{3} - 4 = \frac{8}{3} - \frac{12}{3} = -\frac{4}{3} \] 2. **Calculate \(F(3)\)**: \[ F(3) = \frac{3^3}{3} - 3^2 = \frac{27}{3} - 9 = 9 - 9 = 0 \] 3. **Apply the Fundamental Theorem**: Now we substitute these values into the formula: \[ \int^{2}_{3}(x^2 - 2x)dx = F(2) - F(3) = -\frac{4}{3} - 0 = -\frac{4}{3} \] ### Step 4: Adjust for the Limits of Integration Since we integrated from 3 to 2, we need to reverse the limits, which changes the sign of the result: \[ \int^{2}_{3}(x^2 - 2x)dx = -\left(-\frac{4}{3}\right) = \frac{4}{3} \] ### Conclusion: Final Answer Thus, the value of the integral \(\int^{2}_{3}(x^2 - 2x)dx\) is \(\frac{4}{3}\). ### Explanation of Options - **Option A (4)**: This is incorrect because the area calculated does not equal 4. - **Option B (2)**: This is incorrect as the area calculated does not equal 2. - **Option C (4/3)**: This is the correct answer as derived from our calculations. - **Option D (1/3)**: This is incorrect because the area calculated does not equal 1/3. ### Revision Summary - The integral \(\int^{2}_{3}(x^2 - 2x)dx\) was evaluated using the antiderivative. - The antiderivative of \(x^2 - 2x\) is \(\frac{x^3}{3} - x^2\). - The definite integral was calculated as \(-\frac{4}{3}\) and adjusted for the limits of integration. - The final answer is \(\frac{4}{3}\), corresponding to option C.