Processing math: 100%


SOLUTION 23: Compute the area of the region enclosed by the graphs of the equations y=x,y=3x, and x=2 . Now see the given graph of the enclosed region.

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a.) Using vertical cross-sections to describe this region, we get that 0x2  and  xy3x , so that the area of this region is AREA=20(Top  Bottom) dx =20(3xx) dx =202x dx =(x2)|20 =(2)2(0)2 =4 b.) Using horizontal cross-sections to describe this region, which is made up of two smaller regions, we get that 0y2  and  13yxy in addition to 2y6  and  13yx2 , so that the area of this region is AREA=20(Right  Left) dx+62(Right  Left) dx =20(y13y) dy+62(213y) dy =20(23y) dy+62(213y) dy =(y23)|20+(2yy26)|62 =((2)23(0)23)+((2(6)(6)26)(2(2)(2)26)) =(43)+(6)(423) =4

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