(by integration by parts) y^ y = cos x - T - 1 2 (5 -1, 5) EIN - A file (1 + cos 2x) dx = [x + 2 sin 2x 8 - 7 FIGURE 12 The centroid is (2 7 - 1, g ?...
2 page 564.jpg" />I don't understand how to apply the given number 5 to the centroid formula
564CHAPTER 8 Further Applications of Integrationso Formulas 8 givex - x file xf (x) dx - file x cos x dx= x sin x ]gh - f sin x dx (by integration by parts)y^y = cos x- T - 12(5 -1, 5)EIN- A file (1 + cos 2x) dx = [x + 2 sin 2x 8 - 7FIGURE 12The centroid is (2 7 - 1, g ?) and is shown in Figure 12.If the region A lies between two curves y = f(x) and y = g(x), where f (x) =as illustrated in Figure 13, then the same sort of argument that led to Formulas 8 caC,(x,, 2[f(x,) + 9(x)]used to show that the centroid of R is (x, y), wherey =f(x)R9x=A f. x [f(x) - gf.x)] dxy =g(x)aA S. :[f () - [g(x) ]? } dxXibFIGURE 13(See Exercise 51.)EXAMPLE 6 Find the centroid of the region bounded by the line y = x and theparabola y - x?y4SOLUTION The region is sketched in Figure 14. We take f (x) = x, g(x) = x?, a =y=x-and o = 1 in Formulas 9. First we note that the area of the region isky=x?A=f(x -x2)dx-2- -ThereforeAXFIGURE 14x - A fox[fax) - glx)l dx - + fox(x - x) dx- of wee -x ) dx-6- -;"- A HOP - [WP)dx - + file - x" ) dxThe centroid is (2, ;).
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