Estimate the length of the curve y = (x - 1) ^ 2 on [0,3] using n = 1, 2, 3, 4 -line segments

Question

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

GPT 3.5 · Answer

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.