Question 32.1: The Length of a Line Segment Find the length of the line seg......
The Length of a Line Segment
Find the length of the line segment from (0,0) to (3,4).
You already know that the length of the segment can be calculated by the distance formula:
s=\sqrt{3^{2}+4^{2}}=\sqrt{25}=5\;. Let’s verify this with the arc length formula. The equation of the line segment is y={\frac{4}{3}}x,\ 0\leq x\leq3\,. Because y^{\prime}={\frac{4}{3}}\;, the length of the line segment is
s\;=\;\int_{a}^{b}\sqrt{1+f^{\prime}(x)^{2}}d x\;= \int_{0}^{3}\!\sqrt{1+\left[\frac{4}{3}\right]^{2}}\,d x\ =\ \int_{0}^{3}\!\!\!\sqrt{\left(\frac{3^{2}+4^{2}}{3^{2}}\right)}d x\ =\ \int_{0}^{3}\!\!\frac{5}{3}d x\ =\ \left[\frac{5}{3}x\right]_{0}^{3}\ \ =\ 5\,.
The area of a surface of revolution is based on the arc length formula. The idea is to imagine a piece of arc d s={\sqrt{1+\left[f^{\prime}(x)\right]^{2}\,d x}} rotated about an axis.
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