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

  • question_answer The area bounded by the straight lines \[x=0,x=2\]and the curves \[y={{2}^{x}},y=2x-{{x}^{2}}\]is                                               [AMU 2001]

    A)            \[\frac{4}{3}-\frac{1}{\log 2}\]                                          

    B)            \[\frac{3}{\log 2}+\frac{4}{3}\]

    C)            \[\frac{4}{\log 2}-1\]         

    D)            \[\frac{3}{\log 2}-\frac{4}{3}\]

    Correct Answer: D

    Solution :

               Required area =\[\int_{0}^{2}{[{{2}^{x}}-(2x-{{x}^{2}})]\,dx}\]                                            \[=\left[ \frac{{{2}^{x}}}{\log 2}-{{x}^{2}}+\frac{{{x}^{3}}}{3} \right]_{0}^{2}\]                                                                          \[=\frac{4}{\log 2}-4+\frac{8}{3}-\frac{1}{\log 2}\]\[=\frac{3}{\log 2}-\frac{4}{3}\].

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