Find the mass of a wire bent into the shape of the hyperbola

Find the mass of a wire bent into the shape of the hyperbola xy = 1 from x = 1/2 to x = 2 if its density is given by

p = x^5 / 17

could someone help me

Solution

xy = 1

==>y=1/x

y \'=-1/x²

ds=(1 +(y\')²) dx

ds=(1 +(-1/x²)²) dx

ds=(1/x²)(x⁴+1) dx

mass =_{[1/2 to 2]} p ds

=_{[1/2 to 2]} ( x⁵ /17)(1/x²)(x⁴+1) dx

=(1/17)_{[1/2 to 2]} ( x³)(x⁴+1) dx

let x⁴+1= u==>4x³ dx =du ==>x³ dx =du/4

x=1/2 ==>u =17/16 , x=2 ==> u=17

=(1/17)_{[17/16 to 17]} u du/4

=(1/(4*17))_{[17/16 to 17]} u du

=(1/(4*17))_{[17/16 to 17]} (2/3)u^3/2

=(1/(4*17)) (2/3)[17^3/2-(17/16)^3/2]

=(1/(4*17)) (2/3)[17^3/2(1-(1/16)^3/2)]

= (1/6)[17^1/2(1-(1/16)^3/2)]

= (1/6)[17^1/2(1-(1/64))]

= (1/6)[17^1/2(63/64)]

= (1/6)(63/64)(17^1/2)

= (21/128)(17^1/2)

= (2117)/128