Use n = 3 subdivisions and left endpoints to estimate the area under the graph of
f(x) = 3x
2 + 1 between x = 0 and x = 1.
To estimate the area under the graph of the function f(x) = 3x^2 + 1 between x = 0 and x = 1 using n = 3 subdivisions and left endpoints, you can follow these steps:
1. Divide the interval [0, 1] into n subdivisions. In this case, since n = 3, we will divide the interval into 3 equal subdivisions: [0, 1/3], [1/3, 2/3], and [2/3, 1].
2. Determine the width of each subdivision. Since we have 3 subdivisions, the width of each subdivision will be 1/3.
3. For each subdivision, evaluate the function at the left endpoint. The left endpoints of the subdivisions will be 0, 1/3, and 2/3.
4. Calculate the area of each rectangle. The area of each rectangle can be found by multiplying the width of the subdivision by the height, which is the value of the function at the left endpoint.
- For the first subdivision ([0, 1/3]), the left endpoint is 0. So the height of the rectangle is f(0) = 3(0)^2 + 1 = 1, and the width is 1/3.
- For the second subdivision ([1/3, 2/3]), the left endpoint is 1/3. So the height of the rectangle is f(1/3) = 3(1/3)^2 + 1 = 4/9 + 1 = 13/9, and the width is 1/3.
- For the third subdivision ([2/3, 1]), the left endpoint is 2/3. So the height of the rectangle is f(2/3) = 3(2/3)^2 + 1 = 4/3 + 1 = 7/3, and the width is 1/3.
5. Add up the areas of all the rectangles to get the estimated area under the graph. In this case, we have three rectangles with areas:
- Area of the first rectangle: (1/3) * 1 = 1/3
- Area of the second rectangle: (1/3) * (13/9) = 13/27
- Area of the third rectangle: (1/3) * (7/3) = 7/9
Therefore, the estimated area under the graph using left endpoints and n = 3 subdivisions is (1/3) + (13/27) + (7/9).