What is the arc length of #f(x)=xe^(2x-3) # on #x in [3,4] #?
1 Answer
May 22, 2018
Explanation:
#f(x)=xe^(2x-3)#
#f'(x)=(2x+1)e^(2x-3)#
Arc length is given by:
#L=int_3^4sqrt(1+(f'(x))^2)dx#
Rearrange:
#L=int_3^4f'(x)sqrt(1+(f'(x))^-2)dx#
For
#L=int_3^4f'(x){sum_(n=0)^oo((1/2),(n))(f'(x))^(-2n)}dx#
Isolate the
#L=int_3^4f'(x)dx+sum_(n=1)^oo((1/2),(n))int_3^4(f'(x))^(1-2n)dx#
Hence
#L=f(4)-f(3)+sum_(n=1)^oo((1/2),(n))int_3^4((2x+1)e^(2x-3))^(1-2n)dx#
Apply the substitution
#L=4e^5-3e^3+1/2sum_(n=1)^oo((1/2),(n))int_3^5(e^(4-u)/u)^(2n-1)dx#