question_answer

The area of the region bounded by\[y=-1\],\[y=2,\,\,x={{y}^{3}}\]and\[x=0\]is

A) \[\frac{17}{4}sq\,\,units\]

B)                        \[\frac{1}{4}sq\,\,unit\]

C) \[4\,\,sq\,\,units\]         

D)         None of these

Correct Answer: A

Solution :

Required area\[=\int_{-1}^{2}{|{{x}_{1}}|dy}\]                 \[=\int_{-1}^{0}{(-{{x}_{1}})dy+\int_{0}^{2}{{{x}_{1}}dy}}\]                 \[=-\int_{-1}^{0}{{{y}^{3}}dy+\int_{0}^{2}{{{y}^{3}}dy}}\]                 \[=-\left[ \frac{{{y}^{4}}}{4} \right]_{-1}^{0}+\left[ \frac{{{y}^{4}}}{4} \right]_{0}^{2}=\frac{1}{4}+4\]                 \[=\frac{17}{4}\,\,sq\,\,units\].