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Question
Find the area of the region.
{(x,y) : 0 ≤ y ≤ x2 , 0 ≤ y ≤ x + 2 ,-1 ≤ x ≤ 3} .
Solution
0 ≤ y ≤ x2 ; 0 ≤ y ≤ x + 2 ; -1 ≤ x ≤ 3
y = x2
y = x + 2
x2 = x + 2
x2 - x - 2 = 0
( x - 2 ) ( x + 1) = 0
⇒ x = - 1 , 2
∴ Required area is area of shaded portion
`Delta = int_(-1)^2 (Y_"line" - Y_"parabola" ) dx + int_2^3 Y_"line" dx`
`Delta = int_(-1)^2 ( x + 2 -x^2 ) dx + int_2^3 (x +2 ) dx`
`Delta = int_(-1)^2 [x^2/2 + 2x - x^3/3 ] + int_2^3 [ x^2/2 + 2x]`
`Delta = (2+ 4 - 8/3) - (1/2 - 2 + 1/3) + (9/2 + 6) - (2 + 4 ) `
`Delta = 10/3 + 2/3 +9/2`
`Delta = 4 + 9/2 = 17/2 ` Sq.units
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