A curve passing through (1,2) has its slope at any point (x, y) equal to 2 y-2 . Find the area of the region bounded by the curve and the line 2 x-y-4=0 .

Step-by-step solution

y^ = 2 y-2 y^ y-2 y^ =2 Integrating on both sides gives y^2 2 -2 y=2 x+Cf(1)=2 curve is parabola P: y^2=4(x+y)-8 aligned (y-2)^2 & =4(x-2)+4 Y^2 & =4(X+1) aligned Where X=x-2 and Y=y-2 Line become L: Y=2 X-2 Hence, required area A= _ -2 ^4 ( ( Y+2 2 )- ( Y^2 4 -1 ) ) d y=9