parenrightBigg dy 5 I integraldisplay 1 8 3 8 3 y

# Parenrightbigg dy 5 i integraldisplay 1 8 3 8 3 y

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parenrightBiggdy5.I=integraldisplay18/3parenleftbiggintegraldisplay83y2f(x, y)dxparenrightbiggdyExplanation:The region of integration is similar to theone in the figure89183
byrne (hcb539) – Homework 11 – spice – (54070)3This shaded region is enclosed by the graphsof 9y2=x, y= 1,andx= 8. To changethe order of integration, first fixy.Then,as the graph shows,xvaries from 8 to 9y2.To cover the region of integration, therefore,ymust vary from8/3 to 1.Hence, afterchanging the order of integration,I=integraldisplay18/3parenleftBiggintegraldisplay9y28f(x, y)dxparenrightBiggdy.00610.0pointsReverse the order of integration in the inte-gralI=integraldisplay80integraldisplay4x/2f(x, y)dy dx ,but make no attempt to evaluate either inte-gral.1.I=integraldisplay40integraldisplay8yf(x, y)dx dy2.I=integraldisplay80integraldisplayy4f(x, y)dx dy3.I=integraldisplay80integraldisplay4y/2f(x, y)dx dy4.I=integraldisplay40integraldisplay82yf(x, y)dx dy5.I=integraldisplay80integraldisplayy/20f(x, y)dx dy6.I=integraldisplay40integraldisplay2y0f(x, y)dx dycorrectExplanation:The region of integration is the set of allpointsbraceleftBig(x, y) :x/2y4,0x8bracerightBigin the plane bounded by they-axis and thegraphs ofy=x2,y= 4.This is the shaded region inxy48Integration is taken first with respect toyforfixedxalong the dashed vertical line.To change the order of integration, now fixyand letxvary along the solid horizontal lineinxy48Since the equation of the slant line isx= 2y,integration inxis along the line from (0, y)to (2y, y) for fixedy, and then fromy= 0 toy= 4.Consequently, after changing the order ofintegration,I=integraldisplay40integraldisplay2y0f(x, y)dx dy.keywords: double integral, reverse order inte-gration, linear function,00710.0pointsFind the value of the integralI=integraldisplay integraldisplayA(3(x-3)-4y)dxdy
byrne (hcb539) – Homework 11 – spice – (54070)4whenAis the regionbraceleftBig(x, y) : 0yx-3,3x5bracerightBig.1.I=1032.I= 43.I=83correct4.I= 35.I=113Explanation:The integral can be written as the repeatedintegralI=integraldisplay53parenleftbiggintegraldisplayx-30(3(x-3)-4y)dyparenrightbiggdx.Nowintegraldisplayx-30(3(x-3)-4y)dy=bracketleftBig3(x-3)y-2y2bracketrightBigx-30.ConsequentlyI=integraldisplay53(x-3)2dx=bracketleftBig13(x-3)3bracketrightBig53,and soI=83.00810.0pointsReverse the order of integration in the inte-gralI=integraldisplayln60parenleftBigintegraldisplayex1f(x, y)dyparenrightBigdx ,but make no attempt to evaluate either inte-gral.1.I=integraldisplay60parenleftBigintegraldisplayey6f(x, y)dxparenrightBigdy2.I=integraldisplay60parenleftBigintegraldisplay6eyf(x, y)dxparenrightBigdy3.I=integraldisplayln60parenleftBigintegraldisplayey1f(x, y)dxparenrightBigdy4.I=integraldisplay61parenleftBigintegraldisplayln6lnyf(x, y)dxparenrightBigdycorrect5.I=