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SOLUTION 13: Compute the area of the region enclosed by the graphs of the equations y=x3 and y=4x . Begin by finding the points of intersection of the two graphs. From y=x3 and y=4x we get that x3=4x   x34x=0   x(x24)=0   x(x2)(x+2)=0    x=0,x=2,  or  x=2 Now see the given graph of the enclosed region.

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Using vertical cross-sections to describe this region, which is made up of two smaller regions, we get that 2x0  and  4xyx3 in addition to 0x2  and  x3y4x , so that the area of this region is AREA=02(Top  Bottom) dx+20(Top  Bottom) dx =02(x34x) dx+20(4xx3) dx =(x444x22)|02+(4x22x44)|20 =( ((0)444(0)22)((2)444(2)22) )+( (4(2)22(2)44)(4(0)22(0)44) ) =( (00)(48) )+( (84)(00) ) =4+4 =8

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