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JEE Main & Advanced Mathematics Definite Integration Question Bank Self Evaluation Test - Application of Integrals

  • question_answer
    Let f(x) be a continuous function such that the area bounded by the curve y = f(x), x-axis and the lines x = 0 and x = a is \[\frac{{{a}^{2}}}{2}+\frac{a}{2}\sin a+\frac{\pi }{2}\cos a\], then \[f\left( \frac{\pi }{2} \right)=\]

    A) 1

    B) \[\frac{1}{2}\]

    C) \[\frac{1}{3}\]

    D) None of these

    Correct Answer: B

    Solution :

    [b] We have, \[\int\limits_{0}^{a}{f(x)dx=\frac{{{a}^{2}}}{2}+\frac{a}{2}\sin a+\frac{\pi }{2}\cos a}\] Differentiating w.r.t. a, we get \[f(a)=a+\frac{1}{2}(\sin \,\,a+a\,\,\cos \,\,a)-\frac{\pi }{2}\sin a\] Put \[a=\frac{\pi }{2};f\left( \frac{\pi }{2} \right)=\frac{\pi }{2}+\frac{1}{2}-\frac{\pi }{2}=\frac{1}{2}\]


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