811. Given a geometric figure, one can ask for its centroid, which is - roughly speaking - the average of the points enclosed by the figure. Consider the unit semicircle 0 ≤ y ≤ √1-x², for example. The centroid is usually denoted by coordi- nates (ī,ÿ). The y-axis symmetry of this example makes it clear that = 0, thus only y needs to be calculated. To this end, imagine that a large number M of points has been scattered uniformly throughout the region. The task confronting us is to add up all their y-coordinates and then divide by M. 1 (a) To reduce the labor needed for this long calculation, it is convenient to divide the region into many thin horizontal strips, each of width Ay. Given any one of these strips, it is reasonable to use the same y-value for every one of the points found within the strip. What y-value would you use, and why is this step both convenient and reasonable? (b) Suppose that y = 0.6 represents a horizontal strip. Show that this strip contains about 2(0.8)Ay M of the points, and that it therefore contributes (0.6). -M to the sum. π/2 (c) If the horizontal strips are represented by the values 9₁, 92, 93, ..., and Yn, then the average of the y-coordinates of all the points in all the strips is approximately 2(0.8)Ay π/2 ₁21 AM π/2 2x2y M+y37 π/2 +3/2 2x3 AY M++ Yn= π/2 2xnAy π/2 where each x₂ = √√1-y. Justify this formula. In particular, explain why this sum is an average y-value and is, in fact, a weighted average. Identify the weights and point out which y-values are weighted most heavily. Explain why the sum of all the weights is 1.