Let R be the region bounded by the parabola y = x^2 and the line y = mx , where m > 0 . Let T be the right-angled triangle formed by the x-axis, the line x = m , and the line y = mx . If the area of the region lying inside the triangle T but outside the region R is 9 , then the value of m is equal to

Step-by-step solution

Let the region bounded by the parabola y = x^2 and the line y = mx be R . The points of intersection are given by x^2 = mx x = 0, x = m . The area of region R is: Area (R) = ₀^ m (mx - x^2) dx = [ m x^2 2 - x^3 3 ]₀^ m = m^3 2 - m^3 3 = m^3 6 The triangle T is bounded by y = 0 , x = m , and y = mx . Its vertices are (0,0) , (m,0) , and (m,m^2) . Area (T) = 1 2 base height = 1 2 m m^2 = m^3 2 The area of the region inside T but outside R is: Area (T) - Area (R) = m^3 2 - m^3 6 = m^3 3 Given that this area is 9 : m^3