Discussion
A parametric curve defined by \(x = \cos\left(\frac{\pi u}{2}\right), y = \sin\left(\frac{\pi u}{2}\right)\) in the range \(0 \le u \le 1\) is rotated about the X- axis by 360 degrees. Area of the surface generated is
Answer: C
The surface area is \(2\pi\).
As u runs from 0 to 1, the angle \(\pi u/2\) runs from 0 to 90 degrees, so the curve is a quarter of the unit circle from (1, 0) to (0, 1). Rotating it about the x axis sweeps out a hemisphere of radius 1:
\[ \begin{aligned} S &= \int 2\pi y\, ds = \int_{0}^{1} 2\pi \sin\left(\tfrac{\pi u}{2}\right)\tfrac{\pi}{2}\, du \\ &= 2\pi \end{aligned} \]
which matches half of a sphere's \(4\pi r^{2}\). \(4\pi\) is the whole sphere, and \(\pi\) or \(\pi/2\) would be the area of flat discs or fractions of the hemisphere.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development