Use n = 6 subdivisions and left endpoints to estimate the area under the graph of
f(x) = 2x + 1 between x = 0 and x = 3.
To estimate the area under the graph of f(x) = 2x + 1 between x = 0 and x = 3 using left endpoints and 6 subdivisions, you can follow these steps:
Step 1: Find the width of each subdivision.
To find the width of each subdivision, you need to divide the total interval (x = 3 - x = 0 = 3) by the number of subdivisions (n = 6). In this case, the width of each subdivision would be 3/6 = 0.5.
Step 2: Determine the left endpoints of each subdivision.
Since you're using left endpoints, the left endpoints of each subdivision would be the starting points of each subdivision. In this case, since the starting point is x = 0, the left endpoints would be:
- Interval 1: x = 0
- Interval 2: x = 0.5
- Interval 3: x = 1
- Interval 4: x = 1.5
- Interval 5: x = 2
- Interval 6: x = 2.5
Step 3: Evaluate the function at each left endpoint.
To estimate the area, you need to evaluate the function f(x) = 2x + 1 at each left endpoint calculated in Step 2.
- For Interval 1: f(0) = 2(0) + 1 = 0 + 1 = 1
- For Interval 2: f(0.5) = 2(0.5) + 1 = 1 + 1 = 2
- For Interval 3: f(1) = 2(1) + 1 = 2 + 1 = 3
- For Interval 4: f(1.5) = 2(1.5) + 1 = 3 + 1 = 4
- For Interval 5: f(2) = 2(2) + 1 = 4 + 1 = 5
- For Interval 6: f(2.5) = 2(2.5) + 1 = 5 + 1 = 6
Step 4: Calculate the sum of the areas for each subdivision.
To estimate the area, you need to calculate the sum of the areas for each subdivision. The area of each subdivision is the width (0.5) times the height (the value of f(x) at the left endpoint). So, the sum of the areas would be:
Area = (0.5 * f(0)) + (0.5 * f(0.5)) + (0.5 * f(1)) + (0.5 * f(1.5)) + (0.5 * f(2)) + (0.5 * f(2.5))
= (0.5 * 1) + (0.5 * 2) + (0.5 * 3) + (0.5 * 4) + (0.5 * 5) + (0.5 * 6)
= 0.5 + 1 + 1.5 + 2 + 2.5 + 3
= 10
Therefore, the estimated area under the graph of f(x) = 2x + 1 between x = 0 and x = 3 using left endpoints and 6 subdivisions is 10 square units.