परवलय `y=x^(2)` एवं y=|x| से घिरे क्षेत्र का क्षेत्रफल ज्ञात कीजिए |

Options

A. `(1)/(2)` sq unit
B. `(1)/(3)` sq unit
C. `(2)/(3)` sq unit
D. None of these

Answer

B

Explanation

Firstly, draw the curve `y=x^(2)` and line `y=|x|={{:("x, x "gt0),("0, x "= 0),("-x, x"lt0):}`

The given area is symmetrical about Y-axis.
`therefore` Area OACO = Area ODBO
Required area = 2 (Area of shaded region in the first quadrant only)
`=2int_(0)^(1)(y_(2)-y_(1))dx=2int_(0)^(1)(x-x^(2))dx` [`because` the curve y = |x| lies above the curve `=x^(2)` in (0, 1), so we take `y_(2)=x` and `y_(1)=x^(2)`]
`=2[int_(0)^(1)xdx-int_(0)^(1)x^(2)dx]=2([(x^(2))/(2)]_(0)^(1)-[(x^(3))/(3)]_(0)^(1))`
`=2{((1)/(2)-0)-((1)/(3)-0)}=(1)/(3)` sq unit