Use the arc length formula [3] to find the length of the...

Use the arc length formula $[3]$ to find the length of the curve
$y=2 x-5,-1 \leqslant x \leqslant 3 .$ Check your answer by noting that
the curve is a line segment and calculating its length by the
distance formula.

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Key Concepts

Arc Length Formula

This formula calculates the length of a curve by integrating the square root of the sum of one plus the square of the derivative of the function. It provides a systematic approach for measuring curves defined by functions, making it possible to handle a variety of shapes and forms by converting the geometric problem into an analytical one involving integration over an interval.

Distance Formula

Derived from the Pythagorean theorem, the distance formula computes the straight-line distance between two points in the coordinate plane. This tool is particularly effective for calculating the length of line segments, and it serves as a useful check on more general methods like the arc length formula when the curve in question is a line.

Linear Functions

Linear functions describe straight lines and have a constant rate of change, which makes their derivatives constant. This property simplifies calculations of arc length because the integrand in the arc length formula becomes constant, and it also directly applies to the use of the distance formula, thereby providing consistency between different methods of measuring a line segment.

Transcript

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