The difference of slope between any two points on a continuous elastic curve of a beam is equal to the area under the M / EI curve between these points.
The distance dt along the vertical line through point B is nearly equal to.
(4.5)
Integration of dt between points A and B yield the vertical distance
between the point B and the tangent from point A on the elastic curve. Thus,
(4.6)
since the quantity M/EI represents an infinitesimal area under the M /EI diagram and distance
from that area to point B, the integral on right hand side of Eq. (4.6) can be interpreted as moment of the area under the M/EI diagram between points A and B about point B . This is the second moment area theorem.
If A and B are two points on the deflected shape of a beam, the vertical distance of point B from the tangent drawn to the elastic curve at point A is equal to the moment of bending moment diagram area between the points A and B about the vertical line from point B , divided by EI .
Sign convention used here can be remembered keeping the simply supported beam of Figure 4.1 in mind. A sagging moment is the positive bending moment diagram and has positive area. Slopes are positive if measured in the anti-clockwise direction. Positive deviation
indicates that the point B lies above the tangent from the point A .
Area of M / EI diagram between A and B 
between the point B and the tangent from point A on the elastic curve. Thus, 
from that area to point B, the integral on right hand side of Eq. (4.6) can be interpreted as moment of the area under the M/EI diagram between points A and B about point B . This is the second moment area theorem. 
indicates that the point B lies above the tangent from the point A . 
