Area of the region
`{(x,y) in R^(2):ygesqrt(|x+3|),5ylex+9le15}` is equal to

Options

A. `1/6`
B. `4/3`
C. `3/2`
D. `5/3`

Answer

c

Explanation

We have, `y ge sqrt(|x+3|)`
`implies y^(2) ge |x+3|`
`implies y^(2) ge (x+3) ` for `x ge -3 and y^(2) ge -(x+3)` for `x lt -3`

Required area A is given by
`A=underset(-4) overset(-3)int ((x+9)/(5)-sqrt(-x-3))dx + underset(-3) overset(1)int ((x+9)/(5)-sqrt(x+3))dx`
`impliesA=(1)/(5)underset(-4)overset(-1)int (x+9) dx -underset(-4)overset(-3)int sqrt(-4-3)dx -underset(-3)overset(-1)intsqrt(x+3)dx`
`implies A=[(1)/(5)[(x+9)^(2))/(2)]_(-4)^(1) +(2)/(3)[(-x-3)^(3//2)]_(-4)^(-3)-(2)/(3)[(x+3)^(3//2)]_(-3)^(1)`
`implies A=(1)/(5)(50-(25)/(2))+(2)/(3)(0-1)-(2)/(3)(8-0)=(3)/(2)`