L₄ is calculated using the left endpoints of the subintervals formed by dividing [4,8] into 4 parts, while R₄ uses the right endpoints. Their average is simply the sum of L₄ and R₄ divided by 2.
To find the left endpoint approximation (L₄), right endpoint approximation (R₄), and their average for the definite integral ∫₄₈ (x² + 3)dx with n = 4, we must first divide the interval [4,8] into 4 equal subintervals. Each subinterval has a width of Δx = (8 - 4)/4 = 1.
To calculate L₄, we evaluate the function at the left endpoints of these subintervals - 4, 5, 6, and 7 - and sum the areas of the resulting rectangles.
The formula is L₄ = Τx[(4² + 3) + (5² + 3) + (6² + 3) + (7² + 3)].
Similarly, we find R₄ by evaluating the function at the right endpoints of the subintervals - 5, 6, 7, and 8 - and summing those areas.
So R₄ = Τx[(5² + 3) + (6² + 3) + (7² + 3) + (8² + 3)].
Finally, we calculate the average of L₄ and R₄ by adding them together and dividing by 2. This yields, Average = (L₄ + R₄) / 2