We have a equation of the form y=f(x). To start, we'll find the derivative of the equation.
dxdy=dxd(3x+2)2Using the chain rule, we can solve it to
dxd(3x+2)2=2(3x+2)⋅3=18x+12The length of a curve is given by
L=∫ab1+(dxdy)2dxSubstituting, we get
L=∫141+(18x+12)2dx=∫141+36(3x+2)2dxSubstituting, u=3x+2⟹dxdu=3⟹dx=31du
L=31∫1+36u2duSubstituting, u=6tanv⟹v=arctan(6u)⟹du=6sec2vdv
L=31∫6sec2v1+tan2vdv=181∫sec3v dvThis is a trivial integration solvable by a reduction formula and the standard formula for secx.
∫secnx dx=n−1secn−2xtanx+n−1n−2∫secn−2x dx
∴∫sec3v dv=2secvtanv+21∫secv dv=2secvtanv+21ln(secv+tanv) The final value is (ignoring the constant of integration)
L=1812secvtanv+ln(tanv+secv)=361+36u2⋅6u+ln(6u+1+36u2)=361+36(3x+2)2⋅6(3x+2)+ln(6(3x+2)+1+36(3x+2)2)
Simplifying, and applying the limits, we get
L=[36ln(∣324x2+432x+145+18x+12∣)+(18x+12)324x2+432x+145]14=36ln(30+90184+7057)+6(147057−5901)≈171.0285971744902
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