Area of region
`{(x,y) in R^(2): y ge sqrt(|x+3|,)5y le x + 9 le 15}`

Options

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

Answer

C

Explanation

Here, `{(x,y)in R^(2):y ge sqrt(|x+3|), 5y le (x+9)le15} `
`therefore y ge sqrt(x+3) `
` rArr y ge { (sqrt(x+3)",","when "x ge -3),(sqrt(-x-3)",","when "x le -3):}`
` "or " y^(2) ge { (x+3",","when "x ge -3),(-3-x",","when "x le -3):}`
Shown as

` "Also, " 5y le (x+9)le15 `
`rArr (x+9)ge 5y " and " x le 6 `
Shown as

` therefore ` Required area=Area of trapezium ABCD - Area of ABE under parabola - Area of CDE under parabola
` =(1)/(2)(1+2)(5)-int_(-4)^(-3)sqrt(-(x+3))dx-int_(-3)^(1)sqrt((x+3))dx `
`=(15)/(2)-[((-3-x)^(3//2))/(-(3)/(2))]_(-4)^(-3)-[((x+3)^(3//2))/((3)/(2))]_(-3)^(1) `
` =(15)/(2)+(2)/(3)[0-1]-(2)/(3)[8-0]=(15)/(2)-(2)/(3)-(16)/(3)=(15)/(2)-(18)/(3)=(3)/(2)`