How to plot bending moment diagram from shear force diagram

Question

With conventions that I adopted shown below

I drew roughly the shear force and bending moment diagram of a simply supported beam with concentrated load at the middle. Now while drawing I drew both shear force and bending moment diagram starting from the same end.
1)First from left end

2)Then from the right end

But here how i found out the bending moment was by algebraically adding the areas under the shear force diagram starting from the one end and plotting them along the length of the beam towards the other end. Thus when finding the area under shear force diagram I took area under negative shear force as negative (i.e. negative bending moment).
Is this wrong? Because I am getting positive sign for bending moment when i start drawing from left end and negative sign for bending moment when I start from the right end. Although the shear force diagram is same from both ends.
I have noticed that I didn't use the sign convention I mentioned in the first place while drawing the bending moment. But I have seen in one of the lecture video that you can add up the shear force diagram areas algebraically and get the bending moment diagram easily .[link of the lecture]

NMech · Answer

First of all the bending moment, according to the usual convention (and the one you are presenting here) is positive.
However, the problem originates when you add the area of the shear force. (Probably without realising it), what you are doing is you integrate the shear force over dx $int_0^L V(x)dx$.
I am using the mathematical notation because it is easier to see what's happening.
What happens when you start from the other end (L) is the following:
$$int_L^0 V(x)dx$$
However, when you reverse the integration limits you get:
$$int_0^L V(x)dx = - int_L^0 V(x)dx$$
So when you are starting from right end, essentially what you are doing is that you get a reverse sign. I.e. a positive shear force, creates a negative bending moment, and vice versa.
In order to make that more easily digestible (for engineering students that are allergic to calculus), the following convention is often used:

How to plot bending moment diagram from shear force diagram

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