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  • Need clarity, kindly explain! - Integral Calculus - JEE Main-3

Let f(x)  be a non-­negative continuous function such that the area bounded by the curve y=f(x),\; x-axis  and the ordinates   x=\frac{\pi }{4}\; and\; x=\beta > \frac{\pi }{4}\; is\; \left ( \beta sin\beta +\frac{\pi }{4}cos\beta +\sqrt{2}\beta \right ).Then\; f\left ( \frac{\pi }{2} \right )\; is

  • Option 1)

    \left ( \frac{\pi }{4}-\sqrt{2}+1 \right )\;

  • Option 2)

    \; \; \left ( \frac{\pi }{4}+\sqrt{2}-1 \right )\; \;

  • Option 3)

    \; \left (1- \frac{\pi }{4}+\sqrt{2} \right )\; \;

  • Option 4)

    \; \left (1- \frac{\pi }{4}-\sqrt{2} \right )

 

Answers (1)

best_answer

As we learnt in 

NEWTON LEIBNITZ THEOREM -

\frac {d}{dt}\left ( \int_{f(t)}^{\phi (t))}F(x)dx \right )=F(\phi(t))\phi^{'}(t)-F(f(t))f^{'}(t)

-

 and

Introduction of area under the curve -

The area between the curve y= f(x),x axis and two ordinates at the point  x=a\, and \,x= b\left ( b>a \right ) is given by

A= \int_{a}^{b}f(x)dx=\int_{a}^{b}ydx

- wherein

 

 
Area\: under \:Curve= \int_{\pi/4}^{\beta}f(x)

 

I= \int_{\pi/4}^{\beta}f(x)

Given,
I= \beta \sin \beta+\frac{\pi}{4}\cos \beta+\sqrt{2} \beta, we\:have

\int_{\pi/4}^{\beta} f(x)= \beta \sin \beta+\frac{\pi}{4} \cos \beta+\sqrt{2} \beta-------------------------------- (1)


Differentiating (1) w.r.t    \beta , we get 

f(\beta)\frac{\Delta\beta}{\Delta\beta}- f(\pi/4)\frac{\delta(\pi/4)}{\delta\beta}= \sin \beta+ \beta \cos \beta- \frac{\pi}{4} \sin \beta+\sqrt{2}

\Rightarrow f(\beta)= \sin \beta+ \beta \cos \beta- \frac{\pi}{4} \sin \beta+\sqrt{2}-------------------------- (2)

At  \beta= \frac{\pi}{2}, \:\:f\left ( \frac{\pi}{2}\right )= 1+\frac{\pi}{2}\times0 -\frac{\pi}{4}+\sqrt{2}

f\left ( \frac{\pi}{2}\right )= 1 -\frac{\pi}{4}+\sqrt{2}

 


Option 1)

\left ( \frac{\pi }{4}-\sqrt{2}+1 \right )\;

This is incorrect option

Option 2)

\; \; \left ( \frac{\pi }{4}+\sqrt{2}-1 \right )\; \;

This is incorrect option

Option 3)

\; \left (1- \frac{\pi }{4}+\sqrt{2} \right )\; \;

This is correct option

Option 4)

\; \left (1- \frac{\pi }{4}-\sqrt{2} \right )

This is incorrect option

Posted by

divya.saini

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