To estimate the length of the curve y = sin x on the interval [0, π] using n line segments, we can use the formula for the length of a curve given by:
L = ∫√(1 + (dy/dx)²) dx
For y = sin x, the derivative dy/dx = cos x, so the formula becomes:
L = ∫√(1 + cos² x) dx
For n = 1 line segment, we divide the interval [0, π] into two equal parts: [0, π/2] and [π/2, π]. We calculate the length of the curve for each part using the formula above and add them together.
L₁ = ∫[0, π/2]√(1 + cos² x) dx
L₂ = ∫[π/2, π]√(1 + cos² x) dx
Then, the length of the curve for n = 1 is L = L₁ + L₂.
Similarly, for n = 2 line segments, we divide the interval [0, π] into three equal parts: [0, π/3], [π/3, 2π/3], and [2π/3, π]. We calculate the length of the curve for each part and add them together.
And so on, for n = 3 and n = 4 line segments.
As we increase the number of line segments used to estimate the curve, the length of the curve will approach the true arc length. Using more line segments will result in a more accurate estimation of the curve length. However, as the number of line segments approaches infinity, the approximation becomes exact and matches the true arc length of the curve.
Estimate the length of the curve y = sin x on [0. pi ] using n = 1, 2, 3, 4 line segments
Comment on the arc length of a curve if continued using more and more line segment
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