The shape of a bending-moment diagram (BMD) is dictated by the mathematical order of the load. A useful rule from the beam relations dM/dx = V and dV/dx = −w is that each integration raises the order of the curve by one: a uniformly distributed load (UDL) makes the shear force vary linearly and the bending moment vary as a second-order (quadratic) function of position — that is, parabolic, which is the correct answer.

For a cantilever carrying a UDL of intensity w over its whole length L, take a section at distance x measured from the free end. The only load on the cut-off portion is the UDL acting over length x, whose resultant wx acts at its centroid, x/2 from the section:

• Bending moment: BM(x) = −w·x·(x/2) = −wx²/2 (a quadratic in x).
• At the free end (x = 0): BM = 0.
• At the fixed end (x = L): BM = wL²/2, the maximum value (and the reaction the support must resist).

Because BM depends on x² (not on x to the first power), the diagram is a smooth curve — a parabola — rising from zero at the free end to its peak at the fixed support.

The other shapes correspond to different loadings. A rectangular BMD implies a constant moment (e.g. a pure couple), which a UDL does not produce. A triangular BMD is what a single concentrated (point) load on a cantilever gives, since that makes BM vary linearly (first order) with x. A trapezoidal shape would arise from combined constant-plus-linear moment distributions, not from a pure UDL from the free end. Hence, for a cantilever under a UDL over its full length, the BMD is parabolic.