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### Section 2-1 : Arc Length

1. Set up, but do not evaluate, an integral for the length of $$y = \sqrt {x + 2}$$ , $$1 \le x \le 7$$ using,
1. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx$$
2. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy$$
Solution
2. Set up, but do not evaluate, an integral for the length of $$x = \cos \left( y \right)$$ , $$0 \le x \le \frac{1}{2}$$ using,
1. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx$$
2. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy$$
Solution
3. Determine the length of $$y = 7{\left( {6 + x} \right)^{\frac{3}{2}}}$$ , $$189 \le y \le 875$$. Solution
4. Determine the length of $$x = 4{\left( {3 + y} \right)^2}$$ , $$1 \le y \le 4$$. Solution