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Question

Question: The area of the region bounded by the curves \(y = x ^ { 2 }\) and y =\|x\| is...

The area of the region bounded by the curves y=x2y = x ^ { 2 } and

y =|x| is

A

16\frac { 1 } { 6 }

B

13\frac { 1 } { 3 }

C

56\frac { 5 } { 6 }

D

53\frac { 5 } { 3 }

Answer

13\frac { 1 } { 3 }

Explanation

Solution

Required area =2[01xdx01x2dx]2 \left[ \int _ { 0 } ^ { 1 } x d x - \int _ { 0 } ^ { 1 } x ^ { 2 } d x \right]=2[{x22}01{x33}01]2 \left[ \left\{ \frac { x ^ { 2 } } { 2 } \right\} _ { 0 } ^ { 1 } - \left\{ \frac { x ^ { 3 } } { 3 } \right\} _ { 0 } ^ { 1 } \right]

=2[(120)(130)]2 \left[ \left( \frac { 1 } { 2 } - 0 \right) - \left( \frac { 1 } { 3 } - 0 \right) \right]= 2[1213]=2[16]=132 \left[ \frac { 1 } { 2 } - \frac { 1 } { 3 } \right] = 2 \left[ \frac { 1 } { 6 } \right] = \frac { 1 } { 3 }.