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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 \[=\int_{a-h}^{a}{\pi {{y}^{2}}dx}\]               
    • \[=\pi \int_{a-h}^{a}{({{a}^{2}}-{{x}^{2}})dx}\],  \[(\because {{x}^{2}}+{{y}^{2}}={{a}^{2}})\]     
    • \[=\pi \left[ {{a}^{2}}x-\frac{1}{3}{{x}^{3}} \right]_{a-h}^{a}=\pi \left[ ({{a}^{3}}-\frac{1}{3}{{a}^{3}}-\left\{ {{a}^{2}}(a-h)-\frac{1}{3}{{(a-h)}^{3}} \right\} \right]\]   
    • \[=\pi \left[ \left( {{a}^{3}}-\frac{1}{3}{{a}^{3}} \right)-\left\{ {{a}^{3}}-{{a}^{2}}h-\frac{1}{3}\left( {{a}^{3}}-3{{a}^{2}}h+3a{{h}^{2}}-{{h}^{3}} \right) \right\} \right]\]    
    • \[=\pi \left[ {{a}^{2}}h-{{a}^{2}}h+a{{h}^{2}}-\frac{1}{3}{{h}^{3}} \right]\]\[=\frac{1}{3}\pi {{h}^{2}}(3a-h)\].


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