# Integraldisplay integraldisplay a radicalbig 9 x 2

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integraldisplay integraldisplayAradicalbig9-x2dxdywhenAis the region bounded the graphs ofy=x,x= 3,y= 0.1.I= 82.I= 113.I= 104.I= 9correct5.I= 7Explanation:Since a trigonometric substitution is neededto integrate the functionf(x) =radicalbig9-x2with respect tox, we representIas a re-peated integral, integrating first with respecttoy. Now the region of integration is similarto the shaded region in
zakaria (mmz255) – HW14 – gilbert – (55485)5xy3Thus as a repeated integralI=integraldisplay30parenleftbigg integraldisplayx0radicalbig9-x2dyparenrightbiggdxwhere integrating first with respect toymeansintegrating along the segmentof the linex=d, 0d3, lying inside the shadedregion. Now after integration the inner inte-gral becomesbracketleftBigyradicalbig9-x2bracketrightBigx0=xradicalbig9-x2.Consequently,I=bracketleftbigg-13(9-x2)3/2bracketrightbigg30= 9.keywords:00710.0pointsLocate the points given in polar coordinatesbyPparenleftBig1,12πparenrightBigQparenleftBig2,34πparenrightBig,RparenleftBig4,14πparenrightBig,among24-2-424-2-41.P:Q:R:2.P:Q:R:3.P:Q:R:correct4.P:Q:R:5.P:Q:R:6.P:Q:R:Explanation:To convert from polar coordinates to Carte-sian coordinates we usex=rcosθ ,y=rsinθ .For then the pointsPparenleftBig1,12πparenrightBigQparenleftBig2,34πparenrightBig,RparenleftBig4,14πparenrightBig,correspond toP:Q:R:in Cartesian coordinates.keywords: polar coordinates, Cartesian coor-dinates, change of coordinates,00810.0pointsWhich, if any, ofA.(-4,4π/3),B.(4, π/3),
zakaria (mmz255) – HW14 – gilbert – (55485)
6C.(4,7π/3),are polar coordinates for the point given inCartesian coordinates byP(2,23)?
00910.0pointsWhich one of the following shaded regionsconsists only of points whose polar coordi-nates satisfy the condition-5π4< θ≤ -π8?