If the area of region S = { ( x , y ) : 2 y - y 2 x 2 2 y , x y } is equal to n + 2 n + 1 - n - 1 , then the natural number n is equal to _______ . [2023]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
If the area of region $S = {(x, y) : 2 y - y^{2} \leq x^{2} \leq 2 y, x \geq y}$ is equal to $\frac{n + 2}{n + 1} - \frac{π}{n - 1},$ then the natural number $n$ is equal to _______ . [2023]

Ans.
(5)

Given, $x^{2} + y^{2} - 2 y \geq 0; x^{2} - 2 y \leq 0; x \geq y$

$Area of segment O A B = \frac{90 °}{360 °} \times π {(1)}^{2} - \frac{1}{2} \times 1 \times 1$

$= (\frac{π}{4} - \frac{1}{2})$

Hence, required area $= \int_{0}^{2} (x - \frac{x^{2}}{2}) d x - (\frac{π}{4} - \frac{1}{2})$

$= {(\frac{x^{2}}{2} - \frac{x^{3}}{6})}_{0}^{2} - (\frac{π - 2}{4}) = 2 - \frac{4}{3} - \frac{π}{4} + \frac{1}{2} = \frac{7}{6} - \frac{π}{4}$

Now, $\frac{n + 2}{n + 1} - \frac{π}{n - 1} = \frac{7}{6} - \frac{π}{4}$

For $n = 5$ , $\frac{n + 2}{n + 1} - \frac{π}{n - 1} = \frac{7}{6} - \frac{π}{4}$ $∴$ $n = 5$

Want Us to Email you About Special Offers & Updates?