Let the area of the region {x, y}:2y≤x^2+3, y+|x|≤3, y≥|x-1|} be A.

Question

Let the area of the region \(\left\{(x, y): 2 y \leq x^{2}+3\right., y+|x| \leq 3, y \geq|x-1|\}\) be A. Then 6A is equal to:

(1) 16

(2) 12

(3) 18

(4) 14

Answer 1

Correct option is (4) 14

\( \int_{-1}^1\left(\frac{x^2+3}{2}+(x-1)\right) d x+\int_1^2((3-x)-(x-1)) d x \)

\(\Rightarrow \frac{x^3}{6}+\frac{3 x}{2}+\frac{x^2}{2}-\left.x\right|_{-1} ^1+3 x-\frac{x^2}{2}-\frac{x^2}{2}+\left.x\right|_1 ^2 \)

\(A=\frac{7}{3} \Rightarrow 6 A=14\)