Question 7 of 9intermediate🔧 ApplyNumerical4 marks

Find the area bounded by the curve y=sin⁡xy = \sin x between x=0x = 0 and x=2πx = 2\pi.

Correct Answer

4 square units

Exercise: Miscellaneous Exercise on Chapter 8 | Q: 3 | (Page Page 7)
For More Understanding

Explanation

Following the same method as Example 4 in the textbook for y = cos x, we need to split the region based on where sin x is positive or negative. The curve y = sin x is above x-axis for x ∈ (0, π) and below x-axis for x ∈ (π, 2π). We take absolute value for the region below x-axis to get positive area.

Solution Steps

  1. Step 1: Identify regions where y = sin x is above/below x-axis. For x ∈ (0, π), sin x > 0 (above x-axis). For x ∈ (π, 2π), sin x < 0 (below x-axis).

  2. Step 2: Set up the required area = ∫₀^π sin x dx + |∫_π^{2π} sin x dx|

  3. Step 3: Calculate ∫₀^π sin x dx = [-cos x]₀^π = -cos π - (-cos 0) = -(-1) + 1 = 1 + 1 = 2

  4. Step 4: Calculate ∫_π^{2π} sin x dx = [-cos x]_π^{2π} = -cos 2π - (-cos π) = -1 - 1 = -2. Taking absolute value: |-2| = 2

  5. Step 5: Total required area = 2 + 2 = 4 square units

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