Compute the area of the figure bounded by the straight lines x = 0, x = 2 and the curves y = 2 x , y = 2x –x 2 .
Text Solution
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Sol. The equation y= 2x –x 2 represents a parabola opening downwards having vertex at (1, 1) and crossing x- axis at (0, 0) and (2, 0).
The equation y = 2 x represents the exponential curve as shown in fig. Lines x = 0 and x = 2 are shown in fig. The area bounded by these curves is shaded in fig. We slice the shaded region into vertical strips. For the approximating rectangle shown in fig., we have, Length = (y 1 –y 2 ), width = Δ x
∴ Area = (y 1 –y 2 ) Δ x
The approximating rectangle can move horizontally between x = 0 and x = 2. So,
required area = 
= 
= 
= – 4 +
– 
= 
– 
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