of the lamina of uniform density bounded by the graphs of y=5-x^2 and y=1 Soluti
ID: 3191655 • Letter: O
Question
of the lamina of uniform density bounded by the graphs of y=5-x^2 and y=1Explanation / Answer
ANS 2y - y² = 0 ==> y = 0 or y = 2 ==> (0,0) and (0,2) D = { (x,y) | 0 = x = 2y - y² ; 0 = y = 2 } M = ? ??over D dx dy = ? ?(y = 0 to 2) ?(x = 0 to 2y - y²) dx dy = = ? ?(y = 0 to 2) 2y - y² dy = = ? (y² - y³/3) | (y = 0 to 2) = 4?/3 Mx = ? ??over D y dx dy = ? ?(y = 0 to 2) ?(x = 0 to 2y - y²) y dx dy = = ? ?(y = 0 to 2) 2y² - y³ dy = = ? ( 2y³/3 - y4/4) | (y = 0 to 2) = 4?/3 My = ? ??over D x dx dy = ? ?(y = 0 to 2) ?(x = 0 to 2y - y²) x dx dy = = ? ?(y = 0 to 2) 2y² - 2y³ + y4/2 dy = = ? ( 2y³/3 - y4/2 + y5/10 ) | (y = 0 to 2) = 8?/15 x = My/M = 2/5 y = Mx/M = 1 As you can see, since the function is symmetric about y = 1, that value is the vertical coordinate of the centre of mass and would not require calculation, but I did it in whole scholastic method.
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