Area of an enclosed region bounder by the curve y = f(x), x-axis and the boundaries,x = a to b is given by

A = _a∫^b f(x) dx

Hence, here the area of the triangle ABC is enclosed by the lines AB, BC & CA; its area by integration is given by

Area under AB + Area under BC - Area under AC

iii) Using two point form equation of AB, BC and CA are respectively:

y = (5x - 18)/2, y = (12 - x) and y = (3x - 8)/4

iv) Area under AB = ₄∫⁶ (5x - 18)/2 dx = (1/2[5x² /2 - 18x ₄]⁶
 
=>  AB = (1/2)[(90 - 108) - (40 - 72)] = 7

Similarly, area under BC = 10

and area under AC = 10

Hence required area = 7 + 10 - 10 = 7 sq units.