Set up a Riemann sum to estimate the area under the graph of f(x) = 5x

2 + 2 between
x = 0 and x = 1 using 3 subdivisions and left endpoints. Draw the graph and the 3
rectangles

To set up a Riemann sum, you first need to divide the interval [0, 1] into equal subdivisions. In this case, we have 3 subdivisions, so we can divide the interval as follows:

Subdivision 1: [0, 1/3]
Subdivision 2: [1/3, 2/3]
Subdivision 3: [2/3, 1]

Next, we need to determine the left endpoints of each subdivision. Since we're using left endpoints, we take the value of x at the beginning of each subdivision.

Left endpoint of Subdivision 1: x = 0
Left endpoint of Subdivision 2: x = 1/3
Left endpoint of Subdivision 3: x = 2/3

Now, we can calculate the height for each rectangle by evaluating f(x) = 5x
2 + 2 at the left endpoints.

Height of Rectangle 1: f(0) = 5(0)
2 + 2 = 2
Height of Rectangle 2: f(1/3) = 5(1/3)
2 + 2 = 4
Height of Rectangle 3: f(2/3) = 5(2/3)
2 + 2 = 8/3

Finally, we can draw the graph and the rectangles on a coordinate plane. The x-axis represents the interval [0, 1] and the y-axis represents the values of f(x).

|----------------|----------------|----------------|
| | | |
| | | |
| | | | | | |
| | | | | | |
-----------|--------|-------|--------|-------|--------|-------|---------
Subdiv. 1 Subdiv. 2 Subdiv. 3

In this graph, the rectangles will have a width equal to the width of each subdivision (1/3 in this case). The height of each rectangle is represented by the values we calculated earlier.

Rectangle 1: Width = 1/3, Height = 2
Rectangle 2: Width = 1/3, Height = 4
Rectangle 3: Width = 1/3, Height = 8/3

This setup allows us to estimate the area under the graph of f(x) = 5x
2 + 2 between x = 0 and x = 1 using 3 subdivisions and left endpoints. The estimate can be obtained by calculating the sum of the areas of these rectangles.