Why Is There an Extra Term in the Distributed Load Calculation?

[foo9008]

Homework Statement

For region 8<x<12 , why there's an extra (60-30y +3.75y^2 ) - (30y-7.5y^2) ?
How to get that? I have no idea at all...

Homework Equations

The Attempt at a Solution

here's my working, i have divided the triangle into three parts, i gt 129-80 -75- (60-30y=3.75y^2) - (30y-7.5y^2)-(3.75y^2) instead of 129-80 -75- (60-30y=3.75y^2) - (30y-7.5y^2)
Is the author wrong?

 

[haruspex]

why there's an extra (60-30y +3.75y^2 ) - (30y-7.5y^2)

The aim is to find the net vertical force left of point a, which is y to the left of support B.
The term in question is the contribution from q. The load per unit length from q is 7.5y at a. To avoid confusion, I'll introduce another variable w for a variable distance left of B, ranging from y to 4. The total force over that range from q is therefore $\int _y^47.5w.dw$. If you evaluate that you should arrive at the above expression, though by what logic the author arrived at it I am not sure.