Answer:
A rough sketch of the curve \[x={{y}^{3}}\] is shown below. Clearly, \[y=-\,1\] and y = 2 are Straight lines parallel to X-axis. The required region is shaded in given below figure. 
So, required area A is given by \[A=\int_{-\,1}^{2}{|x|\,dy=\int_{-\,1}^{0}{|x|dy}+\int_{0}^{2}{|x|dy}}\] \[=\int_{-\,1}^{0}{-\,xdy+\int_{0}^{2}{xdy}=\int_{-\,1}^{0}{-\,{{y}^{3}}dy+\int_{0}^{2}{{{y}^{3}}dy}}}\] \[=-\left[ \frac{{{y}^{4}}}{4} \right]_{-1}^{0}+\left[ \frac{{{y}^{4}}}{4} \right]_{0}^{2}=-\left[ 0-\frac{{{(-\,1)}^{4}}}{4} \right]+\left[ \frac{{{2}^{4}}}{4}-0 \right]\] \[=\frac{1}{4}+4=\frac{17}{4}\,\text{sq}\,\,\text{unit}.\]
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