12th Class
Mathematics
Definite Integrals
Question Bank
Critical Thinking
question_answer
The volume of spherical cap of height h cut off from a sphere of radius a is equal to [UPSEAT 2004]
A)\[\frac{\pi }{3}{{h}^{2}}(3a-h)\]
B)\[\pi (a-h)(2{{a}^{2}}-{{h}^{2}}-ah)\]
C)\[\frac{4\pi }{3}{{h}^{3}}\]
D)None of these
Correct Answer:
A
Solution :
The required volume of the segment is generated by revolving the area ABCA of the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] about the x-axis and for the arc BA, Here \[CA=h\] and \[OA=a\], (given)
\ \[OC=OA-CA=a-h\], \ \[x\] varies from \[a-h\] to a.
\ The required volume \[=\int_{a-h}^{a}{\pi {{y}^{2}}dx}\]