A rectangle is formed with one vertex at the origin, opposite vertex on the curve y...

  

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A rectangle is formed with one vertex at the origin, opposite vertex on the curve y = bx", at the point where x = a, (a > 0, b>0, n >0) , and sides parallel to the coordinate axes. Show that the ratio of the area of the part of the rectangle that lies below the curve to the total area of the rectangle depends only on n; the ratio is independent of a and b. 2. (a) Calculate the area of the region in the first quadrant bounded by the coordinate axes and the graph of the parabola y = 1 + a-ax, a > 0. (b) Determine a so that the area found in part (a) is a minimum. A rectangle is formed with one vertex at the origin, opposite vertex on the curve y = bx", at the point where x = a, (a > 0, b>0, n >0) , and sides parallel to the coordinate axes. Show that the ratio of the area of the part of the rectangle that lies below the curve to the total area of the rectangle depends only on n; the ratio is independent of a and b. 2. (a) Calculate the area of the region in the first quadrant bounded by the coordinate axes and the graph of the parabola y = 1 + a-ax, a > 0. (b) Determine a so that the area found in part (a) is a minimum.