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### Section 8.1 : Arc Length

1. Set up, but do not evaluate, an integral for the length of $$y = 14 - 9x$$ , $$- 22 \le y \le 31$$ using,
1. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx$$
2. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy$$
2. Set up, but do not evaluate, an integral for the length of $$x = {{\bf{e}}^{2y}}$$ , $$- 1 \le y \le 0$$ using,
1. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx$$
2. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy$$
3. Set up, but do not evaluate, an integral for the length of $$y = \tan \left( {2x} \right)$$ , $$0 \le x \le \frac{\pi }{3}$$ using,
1. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx$$
2. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy$$
4. Set up, but do not evaluate, an integral for the length of $$\displaystyle \frac{{{x^2}}}{{16}} + 9{y^2} = 1$$.
5. For $$x = 6y + 1$$ , $$- 2 \le y \le 8$$
1. Use an integral to find the length of the curve.
6. Determine the length of $$y = \frac{4}{3}x + 2$$ , $$0 \le x \le 9$$.
7. Determine the length of $$y = {\left( {8x + 3} \right)^{\frac{3}{2}}}$$ , $${11^{\frac{3}{2}}} \le y \le {27^{\frac{3}{2}}}$$.
8. Determine the length of $$x = {\left( {10 - 2y} \right)^{\frac{3}{2}}}$$ , $$- 1 \le y \le 2$$.
9. Determine the length of $$x = 2 + {\left( {y - 1} \right)^2}$$ , $$2 \le y \le 5$$.
10. Determine the length of $$y = {\left( {3x + 2} \right)^2}$$ , $$1 \le x \le 4$$.