The area bounded by the curve y^2=4 a(x-1) and the lines x=1, y=4 a is

Step-by-step solution

The given curve is y^2 = 4a^2(x - 1) . This can be rewritten as x - 1 = y^2 4a^2 , which implies x = y^2 4a^2 + 1 . The area bounded by the curve, the line x = 1 , and the line y = 4a is the area between the curve and the y-axis (or vertical line x=1 ) from y = 0 to y = 4a . The area A is given by the integral A = ₀^ 4a (x - 1) dy . Substituting x - 1 = y^2 4a^2 , we get A = ₀^ 4a y^2 4a^2 dy . Evaluating the integral: A = 1 4a^2 [ y^3 3 ]₀^ 4a . A = 1 4a^2 (4a)^3 3 = 1 4a^2 64a^3 3 . A = 64a^3 12a^2 = 16 3 a^2 . A