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Problem $4-72$:

$1)$ Derive the equations given for beam $3$ in Table A$-9$ using statics and the double$-$integration method. $($Start with the free body diagram, to drive all the way till the ymax$)$

$2)$ Use Castigliano's theorem to verify the maximum deflection for the uniformly loaded cantilever beam $3$ of Appendix Table A$-9.$ Neglect shear