Let's break down the problem step by step: ### 1. **Left Endpoint Estimate (\(L_6\))** - The interval is from \( x=0 \) to \( x=12 \), split into 6 rectangles, so each rectangle has a width of \( \Delta x = \frac{12-0}{6} = 2 \). - The left endpoints are: \( x = 0, 2, 4, 6, 8, 10 \). From the graph, let's estimate the \( y \)-values at these points: - \( f(0) \approx 2 \) - \( f(2) \approx 2 \) - \( f(4) \approx 1.8 \) - \( f(6) \approx 1.5 \) - \( f(8) \approx 1.1 \) - \( f(10) \approx 0.6 \) Calculate \( L_6 \): \[ L_6 = 2 \left[ f(0) + f(2) + f(4) + f(6) + f(8) + f(10) \right] \] \[ L_6 = 2 [2 + 2 + 1.8 + 1.5 + 1.1 + 0.6] = 2 \times 9 = 18 \] ### 2. **Underestimate or Overestimate?** Since the function is **decreasing**, the left endpoint approximation will be an **overestimate** (the rectangles overshoot the curve on the right side). --- ### 3. **Right Endpoint Estimate (\(R_6\))** The right endpoints are: \( x = 2, 4, 6, 8, 10, 12 \). Estimated \( y \)-values: - \( f(2) \approx 2 \) - \( f(4) \approx 1.8 \) - \( f(6) \approx 1.5 \) - \( f(8) \approx 1.1 \) - \( f(10) \approx 0.6 \) - \( f(12) \approx 0 \) Calculate \( R_6 \): \[ R_6 = 2 [2 + 1.8 + 1.5 + 1.1 + 0.6 + 0] = 2 \times 7 = 14 \] ### 4. **Underestimate or Overestimate?** Since the function is **decreasing**, the right endpoint approximation will be an **underestimate** (the rectangles undershoot the curve on the left side). --- ### 5. **Midpoint Estimate (\(M_6\))** The midpoints are: \( x = 1, 3, 5, 7, 9, 11 \). Estimated \( y \)-values: - \( f(1) \approx 2 \) - \( f(3) \approx 1.9 \) - \( f(5) \approx 1.65 \) - \( f(7) \approx 1.3 \) - \( f(9) \approx 0.85 \) - \( f(11) \approx 0.3 \) Calculate \( M_6 \): \[ M_6 = 2 [2 + 1.9 + 1.65 + 1.3 + 0.85 + 0.3] = 2 \times 8 = 16 \] --- ## **Summary of Answers:** 1. \( L_6 = 18 \) 2. Overestimate 3. \( R_6 = 14 \) 4. Underestimate 5. \( M_6 = 16 \) Let me know if you want a more precise estimate for the values!