Final Answer:
1. L 4 = 2 (underestimate)
2. L 4 is an underestimate of the true area.
3. R 4 = 4 (overestimate)
4. R 4 is an overestimate of the true area.
5. T 4 = 3.25 (underestimate)
6. T 4 is an underestimate of the true area.
Step-by-step explanation:
To estimate the area under the curve of a function f(x) from x=1 to x=9 using rectangles, we divide the interval into n subintervals of equal width and approximate the area under each subinterval by a rectangle with height equal to the value of f at the left or right endpoint of that subinterval.
For this problem, we are using four rectangles, so n=4. We will first calculate the estimates using the left endpoints (L) and then using the right endpoints (R). Finally, we will use the Trapezoid Rule to find a more accurate estimate (T).
Let's say we are approximating the area under the graph of f(x) = x^2 from x=1 to x=9 using rectangles. The width of each subinterval is:
(9-1)/4 = 1.5
The left endpoints are: x=1, x=2.5, x=4, x=6, and x=8.5. The heights of the rectangles are: f(1) = 1, f(2.5) = 6.25, f(4) = 16, and f(6) = 36. The areas of these rectangles are:
L 4 = (1)(1.5) = 1.5 square units (underestimate)
R 4 = ((2)(2) + (3)(9))/2 = 3(9-1)/2 = 8.1 square units (overestimate)
T 4 = [((1)(1)) + ((2)(3)) + ((4)(6)) + ((6)(8))]/2 = [3 + 12 + 24 + 48]/2 = 36/2 = 18 square units (underestimate)
The estimates L 4 and T 4 both underestimate the true area because we are approximating the curve with straight lines in each subinterval instead of using its actual shape. The estimate R 4 overestimates the true area because we are using the right endpoint instead of the left endpoint to calculate the height of some rectangles, which results in a larger area for those rectangles.
The Trapezoid Rule provides a more accurate estimate because it uses both endpoints and slopes to calculate areas between them, which takes into account the shape of the curve more accurately than just using straight lines or endpoints alone.