Final answer:
The correct summation to estimate the area under the curve y = 1 + x^2 using 3 rectangles and right endpoints from x = -1 to x = 2 is the sum from i = 1 to 3 of the quantity (i - 1)^2 + 1, which is not represented by any of the given options; hence the answer is 'None of these'.
Step-by-step explanation:
The question is asking us to estimate the area under the curve for y = 1 + x^2 from x = -1 to x = 2 using a Riemann sum with 3 rectangles and right endpoints. To set up the proper summation, we need to divide the range from -1 to 2 into 3 equal intervals. The width of each rectangle (delta x) will be (2 - (-1))/3 = 1. We will evaluate our function at the right endpoints of these intervals, which will be 0, 1, and 2 when starting from -1.
Therefore, the correct summation to estimate the area is:
The summation from i equals 1 to 3 of the quantity (i - 1) squared plus 1
This corresponds to evaluating the function at x = 0, 1, and 2, and summing up the areas of the three rectangles formed thereby.
The given options do not correctly represent these calculations, so the correct answer is None of these.