You want to integrate y=3x-5 from x=-2 to x= 3 ?
This will be the area between the line and the x-axis.
y<0 when 3x-5 < 0 ie x < 5/3
So we work out two areas : one (below x axis) when x < 5/3 (negative area)
One above x-axis (x>=5/3). (positive). Then add these together.
When x = -2, y = -11
When x = 5/3, y = 0.
So what is the area bounded by x-axis, x = -2, x = 5/3.
Well it's triangular. Top side has length 5/3 + 2, left side has length 11. Hopefully you can find that area. (A)
When x = 5/3 , y = 0.
When x = 3. y = 4.
So what is the area between x axis and x=5/3 and x = 3.
Well again we have a triangle. Base = 3 - 5/3, height = 4. Work out the area. (B)
Then to get the integral subtract area A from area B. (We subtract area A because when integrating area below the x axis is negative).