The centroid of a lamina bounded by the graphs of y = f(x) and y = g(x) and the lines x = a and x = b is OP) dx) (x, y) = ( (My Mr) - (-7. (f(x) - 8(x) dx x(f(x)- g(x)) dx Find the centroid of the homogeneous lamina enclosed by the graphs of the equations y=x2/3 and y = x². (Use symbolic notation and fractions where needed. Give your answer as point coordinates in the form (*.*).) (x, y) = (f(x)]²-[g(x)1²) dx (f(x) - g(x)) dx

Minimize f(x,y,z) = x + 2y + 4z subject to the condition ** + y“ +z´ =7 2 II. Find the tangent plane to the graph of **yz +z°e* – y? = 5 at (1,3,2). (10)

2 |3x'e?" dx +5xr'e?" dy C:boundary of the region lying between the squares with vertices (1,1), (-1,1),(-1,-1) and (1,-1) and (3,3), (-3,3), (-3, -3), аnd (3,-3)

What is the additional condition for a cubic-spline S(x) that passes through n data points (x1, Y1), (x2, Y2), .. , (xn, Yn) to be natural? Os" (x1) = 0 and S" (xn) = 0 S (x2) = 0 amd S" (xn-1) = 0 OS' (x1) = 0 and S'(xn) = 0 O (x1) = 0 and S(xn) = 0