How do you find the lengths of the curve #x=(y^4+3)/(6y)# for #3<=y<=8#?
1 Answer
May 19, 2018
Use the arc length formula.
Explanation:
#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#