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Section 8.1 : 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