(Solved): please solve 12 and 18  1. Use the arc length formula (3) to find the length of the curve 17. \ ...

1. Use the arc length formula (3) to find the length of the curve 17. \( y=\ln (\sec x), \quad 0 \leq x \leq \pi / 4 \) \( y=3-2 x,-1 \leqslant x \leqslant 3 \), Check your answer by noting that 18. \( x=e^{y}+\frac{1}{4} e^{-3} . \quad 0 \leqslant y \leqslant 1 \) the curve is a line segment and calculating its length by the distance formula. 19. \( x=\frac{1}{y}(y-3), \quad 1 \leqslant y \leqslant 9 \) 20. \( y=3+1 \cosh 2 x, \quad 0 \leqslant x \leqslant 1 \) 21. \( y=\frac{1}{4} x^{2}-\frac{1}{3} \ln x, \quad 1 \leqslant x \leqslant 2 \) 22. \( y=\sqrt{x-x^{2}}+\sin ^{-1}(\sqrt{x}) \) 23. \( y=\ln \left(1-x^{2}\right), \quad 0 \leq x \leq \frac{1}{7} \) 24. \( y=1-e^{-t}+0 \leq x \leq 2 \) T 27-32 Graph the curve and visually estimate its length. Then compute the length, conrect to four decimal places. 27. \( y-x^{2}+x^{3}, \quad 1 \leqq x \leqslant 2 \) 28. \( y=x+\cos x, \quad 0 \leqslant x \leqslant \pi / 2 \) 3-8 Set up, but do not evaluate, an integral for the length of the curve. 29. \( y=\sqrt[4]{x}, \quad 1 \Leftarrow x \leqslant 4 \) 3. \( y=x^{3}, 0 \leq x \leq 2 \) 4. \( y=e^{2}, 1 \leq x \leq 3 \) 30. \( y=x \tan x, \quad 0 \leqslant x \leqslant 1 \) 5. \( y=x-\ln x, \quad 1 \leqslant x \leq 4 \) 6. \( x=y^{2}+y+\quad 0 \leqslant y \leqslant 3 \) 31. \( y=x e^{-4}, \quad 1 \leq x \leq 2 \) 7. \( x=\sin y, \quad 0 \leqslant y \leqslant \pi / 2 \) 8. \( y^{2}-\ln x, \quad-1 \Leftarrow y \leftrightharpoons 1 \) 32. \( y=\ln \left(x^{2}+4\right), \quad-2 \leq x \leq 2 \) 9-24 Find the exact length of the curve. 33-34 Use Simpson's Rule with \( n-10 \) to estimate the are length of the curve. Compare your answer with the value of the integral produced by a calculator or computer. 9. \( y=\frac{2}{3} x^{3 / 2}, 0 \leqslant x \leqslant 2 \) produced by a calculator of computer. 10. \( y=(x+4)^{v / 2}, 0 \leq x \leq 4 \) 33. \( y=x \sin x, 0 \leqslant x \leqslant 2 \pi \) 34. \( y=e^{-e^{2}}, 0 \leqslant x \leqslant 2 \) 11. \( y=\frac{2}{3}\left(1+x^{2}\right)^{1 / 2}, 0 \leqslant x \leqslant 1 \) 12. \( 36 y^{2}-\left(x^{2}-4\right)^{3}, \quad 2 € x \leqslant 3, y \geqslant 0 \) 35. (a) Graph the curve \( y=x \sqrt{4-x}, 0 \leqslant x \leqslant 4 \). (b) Compute the lengths of approximating polygonal paths 13. \( y=\frac{x^{3}}{3}+\frac{1}{4 x^{2}}, \quad 1 \leqslant x \leqslant 2 \) with \( n=1,2 \), and 4 segments, (Divide the interval into equal subintervals.) Mlustrate by sketching the curve and these paths (as in Figure 6). 14. \( x=\frac{y^{4}}{8}+\frac{1}{4 y^{2}}, \quad 1 \leq y \leq 2 \) (c) Set up an integral for the length of the curve. (d) Compute the length of the curve to four decimal places. Compare with the approximations in part (b). 15. \( y=\frac{1}{2} \ln (\sin 2 x), \quad \pi / 8 \leqslant x \leqslant \pi / 6 \) 36. Repeat Exercise 35 for the curve 16. \( y=\ln (\cos x), \quad 0 \leqslant x \leqslant \pi / 3 \) \( y=x+\sin x \quad 0 \leqslant x \leqslant 2 \pi \)