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The area (sq.units) bounded by the curve y = sinx, π ≤ x ≤ 2π and the x-axis is

Solution

✅ Correct Option: 2

The curve y=sin⁡xy = \sin x for π≤x≤2π\pi \leq x \leq 2\pi lies below the x-axis.

At x=πx = \pi: sin⁡(π)=0\sin(\pi) = 0

At x=3π2x = \frac{3\pi}{2}: sin⁡(3π2)=−1\sin(\frac{3\pi}{2}) = -1

At x=2πx = 2\pi: sin⁡(2π)=0\sin(2\pi) = 0


Since the curve is below the x-axis, ∣sin⁡x∣=−sin⁡x|\sin x| = -\sin x in this interval.

Area =∫π2π∣sin⁡x∣ dx= \int_{\pi}^{2\pi} |\sin x| \, dx

Area =∫π2π(−sin⁡x) dx= \int_{\pi}^{2\pi} (-\sin x) \, dx


The integral of −sin⁡x-\sin x is cos⁡x\cos x.

∫(−sin⁡x) dx=cos⁡x\int (-\sin x) \, dx = \cos x

Applying limits from π\pi to 2π2\pi:

Area =[cos⁡x]π2π= [\cos x]_{\pi}^{2\pi}

Area =cos⁡(2π)−cos⁡(π)= \cos(2\pi) - \cos(\pi)

Area =1−(−1)= 1 - (-1)

Area =2= 2


Therefore, the area bounded by the curve y=sin⁡xy = \sin x and the x-axis for π≤x≤2π\pi \leq x \leq 2\pi is 22 square units.

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