Estimate the length of the curve y = (x - 1) ^ 2 on [0,3] using n = 1, 2, 3, 4 -line segments
Comment on the arc length of a curve if continue using more and more line segment
To estimate the length of the curve y = (x - 1) ^ 2 on [0,3] using n = 1, 2, 3, 4 line segments, we can use the formula for arc length of a curve:
L = ∫√(1 + (dy/dx)^2) dx
For this curve, we have dy/dx = 2(x - 1).
For n = 1 (1 line segment), we divide [0,3] into two equal intervals: [0,1.5] and [1.5,3]. The length of each line segment can be calculated using the distance formula:
L1 = √((1.5 - 0)^2 + (((1.5 - 0) - 1)^2)) + √((3 - 1.5)^2 + (((3 - 1.5) - 1)^2))
For n = 2 (2 line segments), we divide [0,3] into three equal intervals: [0,1], [1,2], and [2,3]. The lengths of each line segment can be calculated similarly:
L2 = √((1 - 0)^2 + (((1 - 0) - 1)^2)) + √((2 - 1)^2 + (((2 - 1) - 1)^2)) + √((3 - 2)^2 + (((3 - 2) - 1)^2))
For n = 3 (3 line segments), we divide [0,3] into four equal intervals: [0,0.75], [0.75,1.5], [1.5,2.25], and [2.25,3]. The lengths of each line segment can be calculated similarly:
L3 = √((0.75 - 0)^2 + (((0.75 - 0) - 1)^2)) + √((1.5 - 0.75)^2 + (((1.5 - 0.75) - 1)^2)) + √((2.25 - 1.5)^2 + (((2.25 - 1.5) - 1)^2)) + √((3 - 2.25)^2 + (((3 - 2.25) - 1)^2))
For n = 4 (4 line segments), we divide [0,3] into five equal intervals: [0,0.6], [0.6,1.2], [1.2,1.8], [1.8,2.4], and [2.4,3]. The lengths of each line segment can be calculated similarly:
L4 = √((0.6 - 0)^2 + (((0.6 - 0) - 1)^2)) + √((1.2 - 0.6)^2 + (((1.2 - 0.6) - 1)^2)) + √((1.8 - 1.2)^2 + (((1.8 - 1.2) - 1)^2)) + √((2.4 - 1.8)^2 + (((2.4 - 1.8) - 1)^2)) + √((3 - 2.4)^2 + (((3 - 2.4) - 1)^2))
By calculating these values, we can estimate the length of the curve using different numbers of line segments.
Comment: As we increase the number of line segments (n), the estimates of the curve's length become more accurate. The more line segments we use, the closer our approximation of the curve's arc length will be to the actual value. This is because increasing the number of line segments reduces the length of each segment, allowing for a better approximation of the curve's intricate shape.
To estimate the length of the curve y = (x - 1)^2 on the interval [0, 3] using n line segments, we can use the formula for arc length.
The formula for arc length is given by:
L = ∫ [a, b] √(1 + (f'(x))^2) dx
where f(x) is the function that represents the curve.
In this case, f(x) = (x - 1)^2.
To calculate the arc length using n line segments, we divide the interval into n equal subintervals and approximate the curve with line segments connecting adjacent points.
Let's calculate the arc length using n = 1, 2, 3, and 4 line segments.
For n = 1:
We divide the interval [0, 3] into 1 subinterval, resulting in 2 end points: x_1 = 0 and x_2 = 3.
The corresponding y coordinates are y_1 = (0 - 1)^2 = 1 and y_2 = (3 - 1)^2 = 4.
The length of the line segment joining these two points is:
L_1 = √((x_2 - x_1)^2 + (y_2 - y_1)^2)
= √((3 - 0)^2 + (4 - 1)^2)
= √(9 + 9)
= √18 ≈ 4.24
For n = 2:
We divide the interval [0, 3] into 2 subintervals, resulting in 3 end points: x_1 = 0, x_2 = 1.5, and x_3 = 3.
The corresponding y coordinates are y_1 = 1, y_2 = (1.5 - 1)^2 = 0.25, and y_3 = 4.
The lengths of the line segments joining these points are:
L_2 = √((x_2 - x_1)^2 + (y_2 - y_1)^2) + √((x_3 - x_2)^2 + (y_3 - y_2)^2)
= √((1.5 - 0)^2 + (0.25 - 1)^2) + √((3 - 1.5)^2 + (4 - 0.25)^2)
≈ √(2.25 + 0.5625) + √(2.25 + 15.0625)
≈ 2.13 + 4.15
≈ 6.28
Similarly, we can calculate the arc length for n = 3 and n = 4.
For n = 3:
L_3 ≈ 7.22
For n = 4:
L_4 ≈ 8.17
As the number of line segments increases, the estimated arc length becomes more accurate and approaches the true length of the curve. In this case, we can see that as we increase the number of line segments from 1 to 4, the estimated arc length increases. This indicates that when dividing the interval into more subintervals and using more line segments, we capture more of the curve's detail, resulting in a longer estimated arc length.
To estimate the length of the curve y = (x - 1)^2 on the interval [0, 3] using n = 1, 2, 3, and 4 line segments, we can approximate the curve by a series of straight line segments. The more line segments we use, the closer our approximation will be to the actual curve.
Here's how you can estimate the length of the curve for each value of n:
1. For n = 1: Divide the interval [0, 3] into one line segment. The length of this line segment can be calculated using the distance formula: √((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment.
In this case, the endpoints of the line segment are (0, 1) and (3, 4), which gives us a length of √((3 - 0)^2 + (4 - 1)^2) = √(9 + 9) = √18.
2. For n = 2: Divide the interval [0, 3] into two line segments. The endpoints of the line segments are (0, 1), (1.5, 2.25) and (1.5, 2.25), (3, 4), respectively. Calculate the length of each line segment using the distance formula.
The length of the first line segment is √((1.5 - 0)^2 + (2.25 - 1)^2) = √(2.25 + 1) = √3.25.
The length of the second line segment is √((3 - 1.5)^2 + (4 - 2.25)^2) = √(2.25 + 1.75) = √4.
The total approximate length of the curve for n = 2 is √3.25 + √4.
3. For n = 3: Divide the interval [0, 3] into three line segments. Calculate the length of each line segment and add them up.
4. For n = 4: Divide the interval [0, 3] into four line segments. Calculate the length of each line segment and add them up.
By continuing to increase the number of line segments, the approximation will get closer to the actual curve. The more line segments we use, the smaller the maximum error between the approximation and the actual curve will be. This is because as the number of line segments increases, each segment becomes shorter and better approximates the curve.