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Question
The area between x-axis and curve y = cos x when 0 ≤ x ≤ 2 π is ___________ .
Options
0
2
3
4
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Solution
` 4`
Required shaded area,
\[A = \int_0^\frac{\pi}{2} \cos x dx + \int_\frac{\pi}{2}^\frac{3\pi}{2} \left( - \cos x \right)dx + \int_{3\frac{\pi}{2}}^{2\pi} \cos x dx\]
\[ = \int_0^\frac{\pi}{2} \cos x dx - \int_\frac{\pi}{2}^\frac{3\pi}{2} \cos x dx + \int_{3\frac{\pi}{2}}^{2\pi} \cos x dx\]
\[ = \left[ \sin x \right]_0^\frac{\pi}{2} - \left[ \sin x \right]_\frac{\pi}{2}^\frac{3\pi}{2} + \left[ \sin x \right]_{3\frac{\pi}{2}}^{2\pi} \]
\[ = \left[ \sin x \right]_0^\frac{\pi}{2} - \left[ \sin x \right]_\frac{\pi}{2}^\frac{3\pi}{2} + \left[ \sin x \right]_{3\frac{\pi}{2}}^{2\pi} \]
\[ = \left( 1 - 0 \right) - \left( - 1 - 1 \right) + \left[ 0 - \left( - 1 \right) \right]\]
\[ = 1 + 2 + 1\]
\[ = 4\text{ sq units }\]
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