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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.

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.

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
ChapterA: Appendix
SectionA.2: Geometric Constructions
Problem 10P: A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in...
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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 (x, y). 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. (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 2(0.8)Ay -M to the sum. π/2 M of the points, and that it therefore contributes (0.6)- π/2 (c) If the horizontal strips are represented by the values y₁, 92, 93, ..., and yn, then the average of the y-coordinates of all the points in all the strips is approximately 4 Y₁ ㅠ 2x2 Ay M+Y3 π/2 √1 − y² dy = - 2x1Ay M+y27 π/2 -1 2x 3 AY M + ... + Yn ² π/2 2x nAy M π/2 where each i = √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. (d) Explain why you can obtain y by calculating y√/1 - y² dy. Evaluate this integral, and check to see that your answer is reasonable. S (e) The integral in part (d) can also be thought of as a weighted average, only now an infinite number of y-values is being averaged. Explain this statement, focusing on the meaning of 2√1-y² π/2 π relates to the weighted average of a finite set of numbers. 1 1) M dy. Evaluate the integral √√1- y² dy and explain how this VI
Transcribed Image Text: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 (x, y). 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. (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 2(0.8)Ay -M to the sum. π/2 M of the points, and that it therefore contributes (0.6)- π/2 (c) If the horizontal strips are represented by the values y₁, 92, 93, ..., and yn, then the average of the y-coordinates of all the points in all the strips is approximately 4 Y₁ ㅠ 2x2 Ay M+Y3 π/2 √1 − y² dy = - 2x1Ay M+y27 π/2 -1 2x 3 AY M + ... + Yn ² π/2 2x nAy M π/2 where each i = √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. (d) Explain why you can obtain y by calculating y√/1 - y² dy. Evaluate this integral, and check to see that your answer is reasonable. S (e) The integral in part (d) can also be thought of as a weighted average, only now an infinite number of y-values is being averaged. Explain this statement, focusing on the meaning of 2√1-y² π/2 π relates to the weighted average of a finite set of numbers. 1 1) M dy. Evaluate the integral √√1- y² dy and explain how this VI
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