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Question

Find the area of the region bounded by y = | x − 1 | and y = 1.

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Solution



We have,y=x-1y=x-1 x11-x x<1
y = x − 1 is a straight line originating from A(1, 0) and making an angle 45o with the x-axis
y = 1 − x is a straight line originating from A(1, 0) and making an angle 135o with the x-axis

y = x is a straight line parallel to x-axis and passing through B(0, 1)

The point of intersection of two lines with y = 1 is obtained by solving the simultaneous equations

y=1and y=x-1 1=x-1x-2=0x=2C2, 1 is point of intersection of y=x-1 and y=1y=1 and y=1-x1=1-xx=0B0, 1 is point of intersection of y=1-x and y=1Since y= x-1 changes character at A(1, 0) ,Consider point P (1,1) on BC such that PA is perpendicular to x-axis.Required shaded area ABCA = area ABPA+area PCAP=011-1-xdx +121-x-1dx=01x dx +122-x dx=x2201+2x-x2212=12+4-2-2+12=12+12=1 sq. unit

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