Area Under The Curves Part 1

Point A is the intersection of y=x+α and y=2−x lines.
∴y=2−y+α
⇒2y=2+α
⇒y=‌

2+α

2

and x=2−‌

2+α

2

=‌

2−α

2

∴ Point A=(‌

2−α

2

,‌

2+α

2

)
∴AN = y-coordinate of point A=‌

2+α

2

Point B is the intersection of y=2+x and y=−x−α lines.
∴y=2−y−α
⇒2y=2−α
⇒y=‌

2−α

2

∴x=‌

2−α

2

−2=‌

2−α−4

2

=‌

−α−2

2

∴ Point B=(‌

−α−2

2

,‌

2−α

2

)
∴BM=y-coordinate of point B=‌

2−α

2

Area of the common region BRAE
=∆CDE−(∆BCR+∆ARD)
=‌

1

2

×4×2−(‌

1

2

(−α+2)×BM+‌

1

2

×(2+α)×AN)
=4−(‌

1

2

×(2−α)×‌

(2−α)

2

+‌

1

2

×(2+α)×‌

2+α

2

)
=4−[‌

(2−α)2

4

+‌

(2+α)2

4

]
Given, 4−[‌

(2−α)2

4

+‌

(2+α)2

4

]=‌

3

2

⇒‌

(2−α)2

4

+‌

(2+α)2

4

=‌

5

2

⇒(2−α)2+(2+α)2=10⇒2α2+8=10
⇒2α2=2
⇒α2=1
⇒α=±1
Given that α>0 so accepted value of α=+1.
Now, 0≤y≤x+2α and |x|≤1
⇒0≤y≤x+2‌ and ‌−1≤x≤1Area of ABCD=‌

1

2

(1+3)×(1−(−1))
=‌

1

2

×4×2
=4 sq. unit