Let the curves be y = 2x and y = |x + 1|. The area

Solution

We are given two curves:

• y = 2x → a straight line passing through origin with slope 2
• y = |x + 1| → a V-shaped curve with vertex at x = –1

We are to find the area between them in the first quadrant. Understand the domain The first quadrant implies:

• x ≥ 0
• y ≥ 0

For x ≥ 0, |x + 1| = x + 1 (since x + 1 ≥ 0) So, in the first quadrant:

• Curve 1: y = 2x
• Curve 2: y = x + 1

Now find the point of intersection in x ≥ 0: Set: 2x = x + 1 ⇒ x = 1 Then y = 2(1) = 2 So the curves intersect at (1, 2) Now consider the region between x = 0 to x = 1 Setting up the definite integral The area A between two curves y₁ (upper) and y₂ (lower) from a to b is: