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University of Toronto at ScarboroughDepartment of Computer & Mathematical SciencesMAT B41H2007/2008Solutions #91.(a)integraldisplay2-3integraldisplayy20(x2+y)dx dy.integraldisplay2-3integraldisplayy20(x2+y)dx dy=integraldisplay2-3bracketleftbiggx33+yxbracketrightbiggy20dy=integraldisplay2-3parenleftbiggy63+y3parenrightbiggdy=bracketleftbiggy721+y44bracketrightbigg2-3=2721+244-parenleftbigg(-3)721+(-3)44parenrightbigg=789584.-32xEqualy2-32(b)integraldisplay1-1integraldisplay(2-y)2y2/3parenleftbigg32x-2yparenrightbiggdx dy.integraldisplay1-1integraldisplay(2-y)2y2/3parenleftbigg32x-2yparenrightbiggdx dy=integraldisplay1-1bracketleftbiggx3/2-2xybracketrightbigg(2-y)2y2/3dy=integraldisplay1-1(2-y)3-2(2-y)2y-vextendsinglevextendsingleyvextendsinglevextendsingle+ 2y5/3dy=-bracketleftbigg(2-y)44bracketrightbigg1-1-2integraldisplay1-14y-4y2+19-11xEqualy2FractionBarExtFractionBarExtFractionBarExt3xEqualLParen12MinusyRParen1219-11y3dy-2integraldisplay10y dy+ 2bracketleftbiggparenleftbigg35parenrightbiggy8/3bracketrightbigg1-1= 20-2bracketleftbigg2y2-43y3+14y4bracketrightbigg1-1-bracketleftbiggy2bracketrightbigg10+ 0 =20-2parenleftbigg-83parenrightbigg-1 =733.(c)integraldisplay1-1integraldisplay1|y|(x+y)2dx dy.The regionvextendsinglevextendsingleyvextendsinglevextendsinglex1,-1y1 can also bedescribed as-xyx, 0x1. So changing theorder of integration we haveintegraldisplay1-1integraldisplay1|y|(x+y)2dx dy=integraldisplay10integraldisplayx-x(x+y)2dy dx=integraldisplay10bracketleftbigg(x+y)33bracketrightbiggx-xdx=13integraldisplay10(2x)3-0dx=83integraldisplay10x3dx=83bracketleftbiggx44bracketrightbigg10=parenleftbigg83parenrightbiggparenleftbigg14parenrightbigg=23.1-11xEqualVertBar1yVertBar1yEqualxyEqualMinusx1-11
MATB41HSolutions # 9page2(d)integraldisplayπ/20integraldisplaycosx0ysinx dy dx.integraldisplayπ/20integraldisplaycosx0ysinx dy dx=integraldisplayπ/20sinxbracketleftbiggy22bracketrightbiggcosx0dx=integraldisplayπ/2012cos2xsinx dx=-16cos3xvextendsinglevextendsinglevextendsinglevextendsingleπ/20= 0 +16=16.ΠFractionBarExtFractionBarExtFractionBarExtFractionBarExt21yEqualCosLBracket1xRBracket1ΠFractionBarExtFractionBarExtFractionBarExtFractionBarExt21(e)integraldisplayD3y dA,whereDis the region bounded byxy2= 1,y=x, x= 0 andy= 3.

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Term
Fall
Professor
moore
Tags
Math, dy dx, dx, dx dy, Ry

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