Let a curve y = y x pass through the point 3 , 3 and the area of the region under this curve, above the x -axis and between the abscissae 3 and x > 3 be y x 3 . If this curve also passes through the point α , 6 10 in the first quadrant, then α is equal to _______.

Step-by-step solution

Given, the area of the region under this curve, above the x -axis and between the abscissae 3 and x > 3 be y x 3 So, ∫ 3 x y d x = y x 3 ⇒ x 3 ∫ 3 x y d x = y 3 Now differentiating both side w.r.t x , we get ⇒ x 3 y + 3 x 2 y 3 x 3 = 3 y 2 d y d x as given  ∫ 3 x y d x = y x 3 ⇒ x 4 + 3 y 2 = 3 y x d y d x ⇒ 3 x y d y d x = 3 y 2 + x 4 Now put y 2 = t , y d y d x = 1 2 d t d x ⇒ d t d x - 2 x t = 2 3 x 3 Now I F = e ∫ - 2 x d x = 1 x 2 So, solution