How do you find the lengths of the curve #x=(y^4+3)/(6y)# for #3<=y<=8#?

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Answer 1 · 2018-05-19T06:32:32

Use the arc length formula.

#x=(y^4+3)/(6y)=y^3/6+1/(2y)#

#x'=1/2(y^2-1/y^2)#

Arc length is given by:

#L=int_3^8sqrt(1+1/4(y^2-1/y^2)^2)dy#

Expand the square:

#L=int_3^8sqrt(1+1/4(y^4-2+1/y^4))dy#

Combine terms:

#L=1/2int_3^8sqrt(y^4+2+1/y^4)dy#

Factorize:

#L=1/2int_3^8sqrt((y^2+1/y^2)^2)dy#

Hence

#L=1/2int_3^8(y^2+1/y^2)dy#

Integrate term by term:

#L=1/2[y^3/3-1/y]_3^8#

Insert the limits of integration:

#L=1/6(8^3-3^3)-1/2(1/8-1/3)#

Simplify:

#L=485/6+5/48=1295/6#