The region bounded by the line y = x, the –axis and the ordinates x = -1, x = 2 is given below :
The area bounded by the line = , the –axis and the ordinates = −1 and = 2 is the area of shaded region = area of the region OCDO + area of the region OABO.
= \(\int\limits_{-1}^{0}\) [0 − y]dx 0 −1 + \(\int\limits^2_0\) yx
( \(\because\) the region OCDO is bounded above by x–axis means y = 0 line and bounded below by line y = x and the region OABO is bounded above by line y = x and bounded below by x–axis )
= \(\int\limits_{-1}^{0}\) ydx + \(\int\limits^2_0\) ydx = \(\int\limits_{-1}^{0}\)ydx + \(\int\limits^2_0\)ydx (\(\because\) \(\int\limits^a_b\)f(x)dx = -\(\int\limits^a_b\)f(x)dx)
Hence, the area bounded by the line y = x , the –axis and the ordinates x = -1, x = 2 is \(\frac{5}{2}\) square units.
