Question 9 of 9advanced🔧 ApplyMCQ1 mark

The area bounded by the curve y=x∣x∣y = x|x|, xx-axis and the ordinates x=−1x = -1 and x=1x = 1 is given by

Exercise: Miscellaneous Exercise on Chapter 8 | Q: 5 | (Page Page 7)
For More Understanding

Explanation

The question asks for the area bounded by y = x|x|, x-axis, and ordinates x = -1 and x = 1. From the context, when the curve is below the x-axis, the area comes out negative, but we take the absolute value. For curves partly above and partly below the x-axis, the total area A = |A₁| + A₂. The function y = x|x| equals x² for x ≥\geq 0 and -x² for x < 0. Thus, from x = -1 to x = 0, the curve lies below the x-axis, and from x = 0 to x = 1, it lies above the x-axis. The area below x-axis (from -1 to 0) gives -1/3, whose absolute value is 1/3. The area above x-axis (from 0 to 1) is 1/3. Total area = 1/3 + 1/3 = 2/3.

Solution Steps

  1. Step 1: Define the function y = x|x|. For x ≥\geq 0, y = x² and for x < 0, y = -x².

  2. Step 2: Identify regions. From x = -1 to x = 0, y = -x² < 0 (below x-axis). From x = 0 to x = 1, y = x² > 0 (above x-axis).

  3. Step 3: Calculate area below x-axis: A₁ = ∫₋₁⁰(-x²)dx = -[x³/3]₋₁⁰ = -[0 - (-1/3)] = -1/3. Taking absolute value: |A₁| = 1/3.

  4. Step 4: Calculate area above x-axis: A₂ = ∫₀¹(x²)dx = [x³/3]₀¹ = 1/3 - 0 = 1/3.

  5. Step 5: Total area = |A₁| + A₂ = 1/3 + 1/3 = 2/3. Hence, option (C) is correct.

Previous