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I am trying to derive the maximum deflection of a cantilever beam. L is the length of the beam and a is the where force is applied.

For the case 0<x<a, I use the method below and it gives me the same result of well known formulas. $$M \left( x \right) ={\it Heaviside} \left( a-x \right) F \left( a-x \right)$$

$$\delta=\int_{0}^{a}\!{\frac {M \left( x \right) \partial \left( M \left( x \right) \right) }{EI\partial \left( F \right) }}\,{\rm d}x $$

For case a<x<L, which method should I follow?

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We calculate the deflection $\delta_a$ and $\theta_a \ \delta_a=\frac{Fa^3}{3EI} \\ \theta_a=\frac{Fa^2}{2EI}$ .

For any point x, a<x<L we add the slope times*(x -a): $\ \theta*(x-a)$ , while noting the slope doesn't chang past point a, along the beam,to the deflection and we get:

$$\delta= \frac{Fa^2}{6EI}(3x-a)$$

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