![\"This](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213255\/CNX_Calc_Figure_06_06_007.jpg\")
Figure 7. A region between two functions.[\/caption]\n\nWe can extend our approach to find centroids of more complex regions. Suppose our region is bounded above by the graph of a continuous function [latex]f(x)[\/latex] and below by a second continuous function [latex]g(x)[\/latex], as shown in the figure.<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"225\"] Figure 7. A region between two functions.[\/caption]\n\n
Again, we partition the interval [latex]\\left[a,b\\right][\/latex] and construct rectangles. A representative rectangle is shown in the following figure.<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"261\"]![\"This](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213257\/CNX_Calc_Figure_06_06_008.jpg\")
Figure 8. A representative rectangle of the region between two functions.[\/caption]\n\n
The centroid of each rectangle is:<\/p>\n
[latex]({x}_{i}^{*},\\frac{f({x}_{i}^{*})+g({x}_{i}^{*})}{2}).[\/latex]<\/p>\n
In the development of the formulas for the mass of the lamina and the moment with respect to the [latex]y[\/latex]-axis, the height of each rectangle is [latex]f(x)\u2212g(x)[\/latex]. For the [latex]x[\/latex]-axis moment, multiply the area by the distance of the centroid from the [latex]x[\/latex]-axis.<\/p>\n
Summarizing these findings, we arrive at the following theorem.<\/p>\n Let [latex]R[\/latex] denote a region bounded above by the graph of a continuous function [latex]f(x),[\/latex] below by the graph of the continuous function [latex]g(x),[\/latex] and on the left and right by the lines [latex]x=a[\/latex] and [latex]x=b,[\/latex] respectively. Let [latex]\\rho [\/latex] denote the density of the associated lamina. Then we can make the following statements:<\/p>\n Let [latex]R[\/latex] be the region bounded above by the graph of the function [latex]f(x)=1-{x}^{2}[\/latex] and below by the graph of the function [latex]g(x)=x-1.[\/latex] Find the centroid of the region.<\/p>\n\n[reveal-answer q=\"fs-id1167793984385\"]Show Solution[\/reveal-answer] The region is depicted in the following figure.<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"273\"] The graphs of the functions intersect at [latex](-2,-3)[\/latex] and [latex](1,0),[\/latex] so we integrate from [latex]\u22122[\/latex] to [latex]1[\/latex]. Once again, for the sake of convenience, assume [latex]\\rho =1.[\/latex]<\/p>\n First, we need to calculate the total mass:<\/p>\n Next, we compute the moments:<\/p>\n and<\/p>\n Therefore, we have<\/p>\n The centroid of the region is [latex](\\text{\u2212}(1\\text{\/}2),\\text{\u2212}(3\\text{\/}5)).[\/latex]<\/p>\n [\/hidden-answer]<\/p>\n<\/section>\n [ohm_question hide_question_numbers=1]288448[\/ohm_question]<\/p>\n<\/section>\n","rendered":" We can extend our approach to find centroids of more complex regions. Suppose our region is bounded above by the graph of a continuous function [latex]f(x)[\/latex] and below by a second continuous function [latex]g(x)[\/latex], as shown in the figure.<\/p>\n Again, we partition the interval [latex]\\left[a,b\\right][\/latex] and construct rectangles. A representative rectangle is shown in the following figure.<\/p>\n The centroid of each rectangle is:<\/p>\n [latex]({x}_{i}^{*},\\frac{f({x}_{i}^{*})+g({x}_{i}^{*})}{2}).[\/latex]<\/p>\n In the development of the formulas for the mass of the lamina and the moment with respect to the [latex]y[\/latex]-axis, the height of each rectangle is [latex]f(x)\u2212g(x)[\/latex]. For the [latex]x[\/latex]-axis moment, multiply the area by the distance of the centroid from the [latex]x[\/latex]-axis.<\/p>\n Summarizing these findings, we arrive at the following theorem.<\/p>\n Let [latex]R[\/latex] denote a region bounded above by the graph of a continuous function [latex]f(x),[\/latex] below by the graph of the continuous function [latex]g(x),[\/latex] and on the left and right by the lines [latex]x=a[\/latex] and [latex]x=b,[\/latex] respectively. Let [latex]\\rho[\/latex] denote the density of the associated lamina. Then we can make the following statements:<\/p>\n Let [latex]R[\/latex] be the region bounded above by the graph of the function [latex]f(x)=1-{x}^{2}[\/latex] and below by the graph of the function [latex]g(x)=x-1.[\/latex] Find the centroid of the region.<\/p>\ncenter of mass of a lamina bounded by two functions<\/h3>\n
\n\t
\n[latex]m=\\rho {\\displaystyle\\int }_{a}^{b}\\left[f(x)-g(x)\\right]dx.[\/latex]<\/div>\n<\/li>\n\t
\n[latex]{M}_{x}=\\rho {\\displaystyle\\int }_{a}^{b}\\frac{1}{2}({\\left[f(x)\\right]}^{2}-{\\left[g(x)\\right]}^{2})dx\\text{ and }{M}_{y}=\\rho {\\displaystyle\\int }_{a}^{b}x\\left[f(x)-g(x)\\right]dx.[\/latex]<\/div>\n<\/li>\n\t
\n[latex]\\overline{x}=\\frac{{M}_{y}}{m}\\text{ and }\\overline{y}=\\frac{{M}_{x}}{m}.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/section>\n
\n[hidden-answer a=\"fs-id1167793984385\"]\n\n![\"This](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213259\/CNX_Calc_Figure_06_06_009.jpg\")
Figure 9. Finding the centroid of a region between two curves.[\/caption]\n\n[latex]\\begin{array}{cc}\\hfill m& =\\rho {\\displaystyle\\int }_{a}^{b}\\left[f(x)-g(x)\\right]dx\\hfill \\\\ & ={\\displaystyle\\int }_{-2}^{1}\\left[1-{x}^{2}-(x-1)\\right]dx={\\displaystyle\\int }_{-2}^{1}(2-{x}^{2}-x)dx\\hfill \\\\ & ={\\left[2x-\\frac{1}{3}{x}^{3}-\\frac{1}{2}{x}^{2}\\right]|}_{-2}^{1}=\\left[2-\\frac{1}{3}-\\frac{1}{2}\\right]-\\left[-4+\\frac{8}{3}-2\\right]=\\frac{9}{2}.\\hfill \\end{array}[\/latex]<\/div>\n[latex]\\begin{array}{cc}\\hfill {M}_{x}& =\\rho {\\displaystyle\\int }_{a}^{b}\\frac{1}{2}({\\left[f(x)\\right]}^{2}-{\\left[g(x)\\right]}^{2})dx\\hfill \\\\ & =\\frac{1}{2}{\\displaystyle\\int }_{-2}^{1}({(1-{x}^{2})}^{2}-{(x-1)}^{2})dx=\\frac{1}{2}{\\displaystyle\\int }_{-2}^{1}({x}^{4}-3{x}^{2}+2x)dx\\hfill \\\\ & =\\frac{1}{2}{\\left[\\frac{{x}^{5}}{5}-{x}^{3}+{x}^{2}\\right]|}_{-2}^{1}=-\\frac{27}{10}\\hfill \\end{array}[\/latex]<\/div>\n[latex]\\begin{array}{cc}\\hfill {M}_{y}& =\\rho {\\displaystyle\\int }_{a}^{b}x\\left[f(x)-g(x)\\right]dx\\hfill \\\\ & ={\\displaystyle\\int }_{-2}^{1}x\\left[(1-{x}^{2})-(x-1)\\right]dx={\\displaystyle\\int }_{-2}^{1}x\\left[2-{x}^{2}-x\\right]dx={\\displaystyle\\int }_{-2}^{1}(2x-{x}^{4}-{x}^{2})dx\\hfill \\\\ & ={\\left[{x}^{2}-\\frac{{x}^{5}}{5}-\\frac{{x}^{3}}{3}\\right]|}_{-2}^{1}=-\\frac{9}{4}.\\hfill \\end{array}[\/latex]<\/div>\n[latex]\\overline{x}=\\frac{{M}_{y}}{m}=-\\frac{9}{4}\u00b7\\frac{2}{9}=-\\frac{1}{2}\\text{ and }\\overline{y}=\\frac{{M}_{x}}{y}=-\\frac{27}{10}\u00b7\\frac{2}{9}=-\\frac{3}{5}.[\/latex]<\/div>\nCenter of Mass of a Region Bounded by Two Functions<\/h2>\n
center of mass of a lamina bounded by two functions<\/h3>\n
\n
[latex]m=\\rho {\\displaystyle\\int }_{a}^{b}\\left[f(x)-g(x)\\right]dx.[\/latex]<\/div>\n<\/li>\n[latex]{M}_{x}=\\rho {\\displaystyle\\int }_{a}^{b}\\frac{1}{2}({\\left[f(x)\\right]}^{2}-{\\left[g(x)\\right]}^{2})dx\\text{ and }{M}_{y}=\\rho {\\displaystyle\\int }_{a}^{b}x\\left[f(x)-g(x)\\right]dx.[\/latex]<\/div>\n<\/li>\n[latex]\\overline{x}=\\frac{{M}_{y}}{m}\\text{ and }\\overline{y}=\\frac{{M}_{x}}{m}.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/section>\n