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    Question

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

    between them in the first quadrant is:
    A 3/2 Correct Answer Incorrect Answer
    B 7/2 Correct Answer Incorrect Answer
    C 2/5 Correct Answer Incorrect Answer
    D 1/2 Correct Answer Incorrect Answer

    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:

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