Arc length is the length of a curve or a segment of a curve in a plane or space. In calculus, the concept of arc length is used to measure the distance along a curve, such as the path of a particle moving through space or the contour of a geometric shape.
To compute the arc length of a curve represented by a function y = f(x) over an interval [a, b], one can use the following integral formula:
arc length = ∫(a to b) √(1 + (f'(x))^2) dx
where f'(x) is the derivative of the function with respect to x. The integral represents the sum of the lengths of infinitesimally small line segments that approximate the curve over the interval [a, b]. As the number of segments approaches infinity, the sum converges to the true arc length of the curve.
For parametric and vector-valued functions, the arc length can be computed using slightly different formulas. For a parametric curve represented by x = f(t) and y = g(t) over an interval [t1, t2], the arc length formula is:
arc length = ∫(t1 to t2) √((dx/dt)^2 + (dy/dt)^2) dt
Similarly, for a vector-valued function r(t) = <f(t), g(t),=”” h(t)=””> in three-dimensional space over an interval [t1, t2], the arc length is given by:</f(t),>
arc length = ∫(t1 to t2) ||r'(t)|| dt
where r'(t) is the derivative of the vector function with respect to the parameter t, and ||r'(t)|| denotes the magnitude of the derivative vector.
Arc length has numerous applications in various fields, including physics, engineering, and computer graphics. It is used to describe the path of a moving object, to measure the perimeter of a two-dimensional shape, and to compute the length of curves in three-dimensional space, such as geodesics on a curved surface.