What is the arclength of #f(x)=(x-3)e^x-xln(x/2)# on #x in [2,3]#?

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Answer 1 · 2018-07-13T17:53:12

#approx 6.533320817#

We are using the Formula

#int_a^bsqrt(1+f'(x)^2)dx#
#f(x)=(x-3)e^x-xln(x/2)#
we Need the first derivative of #f#

#f'(x)=e^x+e^x(x-3)-ln(x/2)-x*(1/(x/2)*1/2)#

simplifying we get

#f'(x)=e^x(x-2)-ln(x/2)-1#
so we have to calculate

#int_2^3sqrt(1+(e^x(x-2)-ln(x/2)-1)^2)dx#
by a numerical method we get

#approx 6.533320817#