Question

What is the formula of arc length of a curve?

Answer 1

The arc length (As) formula for a curve on a circular path is As = Δθ × r, where Δθ is the rotation angle in radians and r is the radius of the circle.

Step-by-step explanation:

The formula to calculate the arc length (As) of a curve, specifically along a circular path, involves the radius of curvature (r) and the angle through which the radius is rotated, often measured in radians. This formula is given by:

As = Δθ × r

where Δθ is the angle of rotation in radians and r is the radius of the circle. For a full revolution, the angle of rotation is 2π radians, since the entire circumference of the circle is covered, which is 2πr.

In summary, the arc length is the product of the angle of rotation and the radius of the circle. This relationship is crucial for understanding circular motion and is often used in problems involving sectors of circles or the analysis of rotational systems.

Answer 2

Arc Length=∫ba√1+[f′(x)]2dx. Note that we are integrating an expression involving f′(x), so we need to be sure f′(x) is integrable. This is why we require f(x) to be smooth.

Step-by-step explanation:

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