JEE Main & Advanced Mathematics Definite Integration Question Bank Area Bounded by Region, Volume and Surface Area of Solids of Revolution

  • question_answer The volume of the solid generated by revolving about the y-axis the figure bounded by the parabola \[y={{x}^{2}}\] and \[x={{y}^{2}}\] is                                                                        [UPSEAT 2002]

    A)            \[\frac{21}{5}\pi \]             

    B)            \[\frac{24}{5}\pi \]

    C)            \[\frac{2}{15}\pi \]             

    D)            \[\frac{5}{24}\pi \]

    Correct Answer: C

    Solution :

               \[V=\int_{0}^{1}{\pi \,{{x}^{2}}dy}\]\[=\pi \int_{0}^{1}{({{y}^{4}}-{{y}^{2}})dy}\]\[=\pi \,\left[ \frac{{{y}^{5}}}{5}-\frac{{{y}^{3}}}{3} \right]_{0}^{1}\]                 \[=\pi \,\left[ \frac{1}{5}-\frac{1}{3} \right]\]\[=\frac{-2\pi }{15}\] i.e., \[\frac{2\pi }{15}\], (Volume should be positive).

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