Class 12

Math

Calculus

Application of Integrals

The area between $x=y_{2}$ and $x=4$ is divided into two equal parts by the line $x=a$, find the value of $a$.

It can be observed that the given area is symmetrical about x-axis.

$⇒$ Area $OED$ $=$ Area $EFCD$

$⇒$ Area $OED$ $=$ Area $EFCD$

Area $OED$ $=∫_{0}ydx$

$=∫_{0}x dx$

$=[23 x_{23} ]_{0}$

$=32 (a)_{23}$ .............. (1)

$=∫_{0}x dx$

$=[23 x_{23} ]_{0}$

$=32 (a)_{23}$ .............. (1)

Area of $EFCD$ $=∫_{a}x dx$

$=[23 x_{23} ]_{a}$

$=[23 x_{23} ]_{a}$

$=32 [8−a_{23}]$ ........... (2)

From (1) and (2), we obtain

$32 (a)_{23}=32 [8−(a)_{23}]$

$⇒2⋅(a)_{23}=8$

$⇒(a)_{23}=4$

$⇒a=(4)_{32}$

$32 (a)_{23}=32 [8−(a)_{23}]$

$⇒2⋅(a)_{23}=8$

$⇒(a)_{23}=4$

$⇒a=(4)_{32}$

Therefore, the value of a is $(4)_{32}$.