Mathematics: Find the length of the curve x equals StartFraction y cubed Over 15 EndFraction plus StartFraction 5 Over 4 y EndFraction on 3 less than or equals y less than or equals 5.

Find the length of the curve x equals StartFraction

Ответ:

carebear147

20.01.2022

Mathematics

6.7

Step-by-step explanation:

The function x(y) can be rewritten as

$x = \frac{y^3}{15} + \frac{5}{4y}$ for $3 \leq y \leq 5$

The integral formula for the curve length is as the following

$L = \int\limits^5_3 {\sqrt{1 + \left(\frac{dx}{dy}\right)^2}} \, dy\\ \frac{dx}{dy} = x' = 0.2y^2 - 1.25y^{-2}\\L = \int\limits^5_3 {\sqrt{1 + (0.2y^2 - 1.25y^{-2})^2}} \, dy\\\\L = \int\limits^5_3 {\sqrt{1 + 0.2^2y^4 - 2*0.2*1.25y^2y^{-2} + 1.25^2y^{-4}}} \, dy\\\\\L = \int\limits^5_3 {\sqrt{0.04y^4 + 0.5 + 1.5625y^{-4}}} \, dy\\\\$