1. **State the problem:** We want to approximate the area under the curve $$y = \frac{5}{2} \sqrt{4 - x^2}$$ on the interval $$0 \leq x \leq 2$$ using a left endpoint Riemann sum with 4 rectangles. 2. **Formula and explanation:** The width of each rectangle is $$\Delta x = \frac{2 - 0}{4} = 0.5$$. For left endpoint Riemann sums, the height of each rectangle is the function value at the left endpoint of each subinterval: $$y_i = f(x_i)$$ where $$x_i$$ are the left endpoints. The area of each rectangle is $$A_i = \Delta x \times y_i$$. 3. **Determine left endpoints:** Since $$\Delta x = 0.5$$, the left endpoints are: $$x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5$$. 4. **Calculate function values at left endpoints:** $$y_0 = \frac{5}{2} \sqrt{4 - 0^2} = \frac{5}{2} \times 2 = 5$$ $$y_1 = \frac{5}{2} \sqrt{4 - 0.5^2} = \frac{5}{2} \sqrt{4 - 0.25} = \frac{5}{2} \sqrt{3.75} = \frac{5}{2} \times 1.936492 = 4.84123$$ $$y_2 = \frac{5}{2} \sqrt{4 - 1^2} = \frac{5}{2} \sqrt{3} = \frac{5}{2} \times 1.732051 = 4.33013$$ $$y_3 = \frac{5}{2} \sqrt{4 - 1.5^2} = \frac{5}{2} \sqrt{4 - 2.25} = \frac{5}{2} \sqrt{1.75} = \frac{5}{2} \times 1.322875 = 3.30719$$ 5. **Calculate areas of rectangles:** $$A_0 = 0.5 \times 5 = 2.5$$ $$A_1 = 0.5 \times 4.84123 = 2.42062$$ $$A_2 = 0.5 \times 4.33013 = 2.16507$$ $$A_3 = 0.5 \times 3.30719 = 1.65360$$ 6. **Sum of areas:** $$\text{Total area} \approx 2.5 + 2.42062 + 2.16507 + 1.65360 = 8.73929$$ **Final answer:** The approximate area under the curve using the left endpoint Riemann sum with 4 rectangles is $$8.73929$$. | x | y | A = \Delta x \times y | |-----|---------|-----------------------| | 0 | 5 | 2.5 | | 0.5 | 4.84123 | 2.42062 | | 1.0 | 4.33013 | 2.16507 | | 1.5 | 3.30719 | 1.65360 | | Sum | | 8.73929 |