The length of the arc formed by the x²=2y³ is 4.5367 unit.
The given arc formed by the x²=2y³
We have to find the length of the arc from point a to point b, where a=(0,0) and b=(4,2).
We can parameterize the curve by sitting y=t, then
x²=2y³
⇒x= √(2y³)
The arc lenght of x²=2y³ parameterized by r(t)= (x(t),y(t))=( √(2y³),t) is given by the integral
[tex]\int\limits^2_0 \sqrt{(\frac{dx}{dt})^2 +(\frac{dy}{dt})^2} \, dx =\int\limits^2_0 \sqrt{\frac{9}{2}t +1 }\, dx[/tex]
[tex]=\frac{2}{9}\int\limits^{10}_1 \sqrt{u }\, dx[/tex] where [tex]u=\frac{9}{2}t +1[/tex]
[tex]=\frac{4}{27}{({u}^{\frac{3}{2}})}^{10}_1\, dx\\=\frac{4(10\sqrt{10}-1) }{27}\\=4.5367[/tex]
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