To approximate the integral using left endpoints and with n = 4, we first need to determine the width of each rectangle. To do this, we subtract the lower limit of the integral (a = 1) from the upper limit of the integral (b = 3) and divide by the number of rectangles (n = 4), giving us a width of 0.5.
We then need to find the left endpoint x-values for each rectangle. Our first rectangle starts at a (1.0), and each subsequent rectangle starts 0.5 units further than the one before it, leading to the left endpoint x-values of 1.0, 1.5, 2.0, and 2.5.
The area of each rectangle is calculated by evaluating the integrand function at the left endpoint x-value and multiplying by the width (0.5). Using this, we get the areas of the first, second, third, and fourth rectangles to be 1.5, 3.0, 5.0, and 7.5 respectively.
Finally, to approximate the integral, we sum these areas together, resulting in an integral approximation of 17.0.
So, based on these calculations, the approximation of the integral ∫ from 1 to 3 of (2x^2 + x) dx using left endpoints and n = 4 is 17.0.