Let S be the region bounded by the curve y = 4 - x^2 and the x-axis. The normal to the curve at the point (1, 3) divides S into two regions. If the area of the smaller region is A , then the value of 48A is equal to

Step-by-step solution

The region S is bounded by the parabola y = 4 - x^2 and the x-axis ( y = 0 ). The total area of S is: Area (S) = _ -2 ² (4 - x^2) dx = [ 4x - x^3 3 ]_ -2 ² = ( 8 - 8 3 ) - ( -8 + 8 3 ) = 32 3 = 512 48 To find the equation of the normal at (1, 3) , we first find the derivative of the curve: dy dx = -2x At x = 1 , the slope of the tangent is m_t = -2 . The slope of the normal is m_n = -1 m_t = 1 2 . The equation of the normal line is: y - 3 = 1 2 (x - 1) y = x 2 + 5 2 Next, we find the points of intersection of the n