The value of int(1,10) f(x) is equal to the sum of signed areas of the region bounded by f(x).
From 1 to 6, the bounded region is a trapezoid: Its bases are (6-1)=5 and (4-3)=1, and its height is 2. So the signed area of the trapezoid is (1/2)*(5+1)*2=6 recalling that the area formula for a trapezoid is 1/2(a+b)h; the signed area is positive in this part is because it is above the x-axis.
From 6 to 10, the bounded region is a triangle: Its base is (10-6)=4 and its height is 2. Since it is below the x-axis, then its signed area is negative, that is, -(1/2)*(4)*(2)=-4.
Therefore, the value of int(1,10) f(x) = 6+(-4) =2.
