Question 4 of 9intermediate🔧 ApplyMCQ1 mark

Area of the region bounded by the curve y2=4xy^2 = 4x, yy-axis and the line y=3y = 3 is

Exercise: EXERCISE 8.1 | Q: 4 | (Page Page 5)
For More Understanding

Explanation

The question asks for the area bounded by the parabola y² = 4x, the y-axis (x = 0), and the horizontal line y = 3. Since the boundaries involve the y-axis and a horizontal line, we use the formula for area with respect to y: A = ∫x dy. From y² = 4x, we get x = y²/4. The region extends from y = 0 (where the parabola meets the y-axis at origin) to y = 3. Integrating x = y²/4 from y = 0 to y = 3 gives us the required area.

Solution Steps

  1. Step 1: Rewrite the curve equation. From y² = 4x, we get x = y²/4

  2. Step 2: Identify the limits of integration. The region is bounded by y-axis (x = 0) and line y = 3. The parabola meets y-axis at origin (0, 0), so limits are y = 0 to y = 3

  3. Step 3: Apply the formula for area with respect to y: A = ∫₀³ x dy = ∫₀³ (y²/4) dy

  4. Step 4: Calculate: A = (1/4) × [y³/3]₀³ = (1/4) × [(27/3) - 0] = (1/4) × 9 = 9/4

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