Let's say that the two curves are f(x,y)=x+y−3=0 and g(x,y)=x2−y−9=0. The points of their intersection are the points where f(x,y)=g(x,y) ⇒x+y−3=x2−y−9=0 ⇒−x−y+3=x2−y−9=0 ⇒x2+x−12=0 ⇒x2+4x−3x−12=0 ⇒x(x+4)−3(x+4)=0 ⇒(x+4)(x−3)=0 ⇒x+4=0 OR x−3=0 ⇒x=−4 OR x=3. And, y=3−(−4)=7 OR y=3−3=0. Hence, the curves intersect at the points B(3, 0) and C(-4, 7) as shown in the diagram below:
The points where y=x2−9 cuts the x -axis (y=0) are A(−3,0) and B(3,0)The required area is the shaded part ABC= Area of BDC - Area of ADC. =