Surely. We will follow an important concept of calculus known as Lower Sums to approximate the area under a curve using rectangles. For our function `f(x) = x^2 + 5` and the range `x = 0` to `x = 3`, let's employ six rectangles to illustrate this process.
First, we establish the width (`h`) of each rectangle. This is determined by subtracting the beginning point of the range (`a = 0`) from the end point of the range (`b = 3`) and then dividing by the number of rectangles (`n = 6`), which gives us `h = (3-0)/6 = 0.5`. Therefore, each rectangle has a width of 0.5.
Second, we determine the x-values at the left side of each rectangle. These are 0, 0.5, 1.0, 1.5, 2.0, and 2.5. They represent the beginning of each of the six rectangles.
Third, we find out the height of each rectangle. This is calculated by plugging our x-values into our function `f(x)`. Our function values, or ‘heights’, at these points are 5, 5.25, 6, 7.25, 9, and 11.25 respectively.
Next, we calculate the area of each rectangle. The area of a rectangle is given by width multiplied by height. Since our width is 0.5 for every rectangle and our heights are the function values we calculated for each rectangle, we just have to multiply them to find the area of each rectangle. The areas are 2.5, 2.625, 3, 3.625, 4.5, and 5.625 respectively.
Lastly, we sum up the areas of all the rectangles to get the approximated area under the curve. The sum of the areas of the rectangles is 21.875.
To conclude, by employing six rectangles, the lower sum estimate for the area under the curve `f(x) = x^2 + 5` on the interval from 0 to 3 is 21.875. Please remember however that this is an estimation. The exact area could be calculated using integral calculus.