Area of the region bounded by the curve \( y=\cos x \) and x-axis between \( x=0 \) and \( x=\pi \) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 2 Sq units
\( \cos x\ge 0 \) on \( \left[0,\frac{\pi}{2}\right] \) and \( \cos x\le 0 \) on \( \left[\frac{\pi}{2},\pi\right] \). Area is always positive, so split at x = π/2 and take the absolute value of the second part:
\( \displaystyle \text{Area}=\int_{0}^{\frac{\pi}{2}}\cos x\,dx+\left|\int_{\frac{\pi}{2}}^{\pi}\cos x\,dx\right| \)
\( \displaystyle =\left[\sin x\right]_{0}^{\frac{\pi}{2}}+\left|\left[\sin x\right]_{\frac{\pi}{2}}^{\pi}\right| \)
\( \displaystyle =(1-0)+|0-1| \)
\( \displaystyle =2\ \text{sq units} \)
Tip: Integrating straight across gives \( \int_{0}^{\pi}\cos x\,dx=0 \), not the area; whenever the curve crosses the x-axis inside the interval, split at the crossing.
NCERT Exemplar:
Application of Integrals
