Show All Notes Hide All Notes

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.
   2. Verify your answer from part (a) geometrically.
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 \(y = {\left( {3x + 2} \right)^2}\) , \(1 \le x \le 4\).