Let’s solve each part step by step for \( f(x) = x^2 - 4x \) on \( 0 \leq x \leq 2 \): ### **Step 1: Partition the Interval** Interval: \([0,2]\), \( n = 4 \) \[ \Delta x = \frac{2 - 0}{4} = 0.5 \] The subintervals are: \( [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2] \) --- ### **Step 2: Right Endpoints, \( n = 4 \) Riemann Sum** Right endpoints: \( x_1 = 0.5, x_2 = 1, x_3 = 1.5, x_4 = 2 \) \[ \text{Riemann sum} = \sum_{i=1}^{4} f(x_i) \Delta x \] Calculate each \( f(x_i) \): - \( f(0.5) = (0.5)^2 - 4(0.5) = 0.25 - 2 = -1.75 \) - \( f(1) = 1 - 4 = -3 \) - \( f(1.5) = (1.5)^2 - 4(1.5) = 2.25 - 6 = -3.75 \) - \( f(2) = 4 - 8 = -4 \) Sum: \[ \text{Sum} = \left[ -1.75 + (-3) + (-3.75) + (-4) \right] \times 0.5 = (-12.5) \times 0.5 = -6.25 \] --- ### **Step 3: Left Endpoints, \( n = 4 \) Riemann Sum** Left endpoints: \( x_0 = 0, x_1 = 0.5, x_2 = 1, x_3 = 1.5 \) \[ \text{Riemann sum} = \sum_{i=0}^{3} f(x_i) \Delta x \] Calculate each \( f(x_i) \): - \( f(0) = 0^2 - 4(0) = 0 \) - \( f(0.5) = -1.75 \) (already calculated) - \( f(1) = -3 \) - \( f(1.5) = -3.75 \) Sum: \[ \text{Sum} = [0 + (-1.75) + (-3) + (-3.75)] \times 0.5 = (-8.5) \times 0.5 = -4.25 \] --- ### **Step 4: Midpoints, \( n = 4 \) Riemann Sum** Midpoints: \( m_1 = 0.25, m_2 = 0.75, m_3 = 1.25, m_4 = 1.75 \) Calculate \( f(m_i) \): - \( f(0.25) = (0.25)^2 - 4(0.25) = 0.0625 - 1 = -0.9375 \) - \( f(0.75) = (0.75)^2 - 4(0.75) = 0.5625 - 3 = -2.4375 \) - \( f(1.25) = (1.25)^2 - 4(1.25) = 1.5625 - 5 = -3.4375 \) - \( f(1.75) = (1.75)^2 - 4(1.75) = 3.0625 - 7 = -3.9375 \) Sum: \[ \text{Sum} = [-0.9375 + (-2.4375) + (-3.4375) + (-3.9375)] \times 0.5 = (-10.75) \times 0.5 = -5.375 \] --- ### **Step 5: Limit of Riemann Sum (Definite Integral Definition)** \[ \int_0^2 (x^2 - 4x) \, dx \] Let’s use the definition: \[ \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x \] But let's calculate exactly: \[ \int_0^2 (x^2 - 4x) dx \] --- ### **Step 6: Fundamental Theorem of Calculus** Find an antiderivative \( F(x) \): \[ F(x) = \int (x^2 - 4x) dx = \frac{1}{3}x^3 - 2x^2 + C \] Evaluate from 0 to 2: \[ F(2) = \frac{1}{3}(8) - 2(4) = \frac{8}{3} - 8 = -\frac{16}{3} \] \[ F(0) = 0 - 0 = 0 \] \[ \int_0^2 (x^2 - 4x)dx = F(2) - F(0) = -\frac{16}{3} - 0 = -\frac{16}{3} \] --- ## **Final Answers** 1. **Right endpoints, \( n = 4 \):** \(\boxed{-6.25}\) 2. **Left endpoints, \( n = 4 \):** \(\boxed{-4.25}\) 3. **Midpoints, \( n = 4 \):** \(\boxed{-5.375}\) 4. **Limit of Riemann sum (definite integral):** \(\boxed{-\frac{16}{3}}\) 5. **Fundamental theorem:** \(\boxed{-\frac{16}{3}}\) Let me know if you need more explanation on any step!