Bending moment:

Let the bending moment (M) be a sagging moment and the M/EI   diagram is drawn (Figure 9). (Here, sagging moments and downward deflections are taken because they are considered positive as per our sign convention. However, the following discussion is true for any deflection or moment with any sign.)

Let the distance AB be 'a' and imagine a vertical line SS′ passing through another point S which is at a distance 's' from A. Coordinate origin is at A and x is measured positive to right and y is positive downwards. θ _A , y _A and θ_B , y_B  are the slopes and deflection at the points A and B respectively. The tangential intercepts on the vertical line SS′ are s tan θ_A and (s - a) tan θB, which are approximately equal to s θ_A and (s - a) θ_B respectively, because the slopes θ_A and θ_B are very small and for such small angles the value of tan θ_A is equal to θ_A (in radians).

Now, from the study of 'Strength of Materials', we know that

dθ /dx= d²y/dx² = - M/EI

The negative sign is taken because for a positive (sagging) moment the slope θ of the beam goes on decreasing as x increases. Integrating the above equation among A and B, we get the following relation:

∫^B_A d θ =-∫^B_A M/EI dx

θB- θA =-∫^B_A M/EI dx

θA- θB =-∫^B_A M/EI dx