Question

find the centre of curvature of the curve x^2/3 + y^2/3 =a^2/3 at any point (x,y)​

Answer 1

Step-by-step explanation:

We can do this by multiple integration.

In multiple integration we used find Centre of mass. The same equation would do.

So Xcm=2 integral (x dx dy)/2 integral(dx dy)

With limits x limits 0 to (a^2/3-y^2/3)^3/2

y limits 0 to a.

We can solve the integral by giving y=a sin^3(t) substitution.

Then we can solve the resulting integral by beta function.

For X centre , I got 4a/21. Did'nt check the steps twice , I believe the procedure is correct.