What is the arc length of #f(x) = (x^2-1)^(3/2) # on #x in [1,3] #?
1 Answer
A first-order approximation gives
Explanation:
#f(x)=(x^2-1)^(3/2)#
#f'(x)=3xsqrt(x^2-1)#
Arc length is given by:
#L=int_1^3sqrt(1+9x^2(x^2-1))dx#
Expand:
#L=int_1^3sqrt(9x^4-9x^2+1)dx#
Complete the square:
#L=1/2int_1^3sqrt(9(2x^2-1)^2-5)dx#
Factorize:
#L=3/2int_1^3(2x^2-1)sqrt(1-5/(9(2x^2-1)^2))dx#
For
#L=3/2int_1^3(2x^2-1){sum_(n=0)^oo((1/2),(n))(-5/(9(2x^2-1)^2))^n}dx#
Isolate the
#L=3/2int_1^3(2x^2-1)dx+3/2sum_(n=1)^oo((1/2),(n))(-5/9)^nint_1^3(1/(2x^2-1))^(2n-1)dx#
Apply the difference of squares:
#L=3/2[2/3x^3-x]_ 1^3+3/2sum_(n=1)^oo((1/2),(n))(-5/9)^nint_1^3 (1/((sqrt2x-1)(sqrt2x+1)))^(2n-1)dx#
Apply partial fraction decomposition:
#L=23+3sum_(n=1)^oo((1/2),(n))(-5/36)^nint_1^3 (1/(sqrt2x-1)-1/(sqrt2x+1))^(2n-1)dx#
Isolate the
#L=23-5/24int_1^3 (1/(sqrt2x-1)-1/(sqrt2x+1))dx+3sum_(n=2)^oo((1/2),(n))(-5/36)^nint_1^3 (1/(sqrt2x-1)-1/(sqrt2x+1))^(2n-1)dx#
Hence
#L=23-5/(24sqrt2)[ln|sqrt2x-1|-ln|sqrt2x+1|]_ 1^3+3sum_(n=2)^oo((1/2),(n))(-5/36)^nint_1^3 (1/(sqrt2x-1)-1/(sqrt2x+1))^(2n-1)dx#
Giving:
#L=23-5/(24sqrt2)ln((5+2sqrt2)/(5-2sqrt2))+3sum_(n=2)^oo((1/2),(n))(-5/36)^nint_1^3 (1/(sqrt2x-1)-1/(sqrt2x+1))^(2n-1)dx#