The area of the region enclosed by the curve f(x)=max\{... | Area of Bounded Region (Mathematics) | DivineJEE

JEE Mains

Mathematics

The area of the region enclosed by the curve

f(x)=max\{sin⁡x,cos⁡x\},- π ≤x ≤\nπ

and the x-axis is

4(\sqrt2)

2(\sqrt2+1)

2\sqrt2(\sqrt2+1)

Correct Answer:

(d): We have

f(x) = \max\{\sin x, \cos x\}, \quad -\pi \le x \le \pi

Now, the area of the region enclosed by the given curve and

x

-axis

= \left| \int_{-\pi}^{-3\pi/4} \sin x \, dx \right| + \left| \int_{-3\pi/4}^{-\pi/2} \cos x \, dx \right| + \int_{-\pi/2}^{\pi/4} \cos x \, dx + \int_{\pi/4}^{\pi} \sin x \, dx

= \frac{-1}{\sqrt{2}} + 1 + 1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} + 1 + 1 + \frac{1}{\sqrt{2}} = 4

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