Question

steps please

Find the arc length of the curve 24 xy = y4 + 48 from y = 2 to y = 4.

Answer 1

24xy = y^4 + 48 => x = (y^4 + 48)/24y

Arc length of some curve x = f(y) on the interval [a,b] = ∫√(1 + (dx/dy)^2) dy from a to b

In this case => dx/dy = (y^2)/8 - 2/(y^2)

Arc length = ∫√(1 + ((y^4)/64 - 1/2 + 4/(y^4)) dy from 2 to 4

= ∫√(32y^4 + y^8 + 256)/(8y^2) dy from 2 to 4

Note that y^8 + 32y^4 + 256 = (y^4 + 16)^2.

Therefore:

∫(y^4 + 16)/8y^2 dy from 2 to 4

= (y^3)/24 - 2/y eval. from 2 to 4

= 8/3 - 1/2 - 1/3 + 1

= 7/3 + 1/2 = 17/6