Suppose that we want to estimate the area under the graph of f(x)=X^2+x for x=1 and 3. What is the value of the estimate using four rectangles and taking the sample points to be left-hand endpoints?

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To estimate the area under the graph of the function f(x) = x^2 + x for x = 1 and 3 using four rectangles and left-hand endpoints, we need to divide the interval [1, 3] into four equal subintervals and approximate the area of each rectangle.

Here's how we can do it step-by-step:

1. Determine the width of each subinterval:
The interval [1, 3] is divided into four equal subintervals, so the width of each subinterval is (3 - 1) / 4 = 0.5.

2. Determine the left-hand endpoints of each subinterval:
For the left-hand endpoints, start with the leftmost point of the interval, which is x = 1. Then add 0.5 to obtain the remaining three left-hand endpoints.
The left-hand endpoints for the four subintervals are: 1, 1.5, 2, and 2.5.

3. Evaluate the function at each left-hand endpoint:
Substitute each of the left-hand endpoints into the function f(x) = x^2 + x to calculate the function values at each point.
f(1) = 1^2 + 1 = 2
f(1.5) = (1.5)^2 + 1.5 = 3.75
f(2) = 2^2 + 2 = 6
f(2.5) = (2.5)^2 + 2.5 = 9.25

4. Calculate the area of each rectangle:
The area of each rectangle is the product of the width (0.5) and the corresponding function value.
Area of the first rectangle = 0.5 * 2 = 1
Area of the second rectangle = 0.5 * 3.75 = 1.875
Area of the third rectangle = 0.5 * 6 = 3
Area of the fourth rectangle = 0.5 * 9.25 = 4.625

5. Sum up the areas of all rectangles to estimate the total area:
Add up the areas of all four rectangles to get the estimate of the total area under the graph.
Total estimated area = 1 + 1.875 + 3 + 4.625 = 10.5 square units.

Therefore, the estimated area under the graph of f(x) = x^2 + x for x = 1 and 3, using four rectangles and left-hand endpoints, is 10.5 square units.