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we have
![= 2\left [ \int_{0}^{2a}\sqrt{6a}\sqrt{xdx} +\int_{2a}^{4a}\sqrt{\left ( 4a \right )^{2}-x^{2}dx}\right ]](https://entrancecorner.codecogs.com/gif.latex?%3D%202%5Cleft%20%5B%20%5Cint_%7B0%7D%5E%7B2a%7D%5Csqrt%7B6a%7D%5Csqrt%7Bxdx%7D%20+%5Cint_%7B2a%7D%5E%7B4a%7D%5Csqrt%7B%5Cleft%20%28%204a%20%5Cright%20%29%5E%7B2%7D-x%5E%7B2%7Ddx%7D%5Cright%20%5D)
![= 2\left [ \left ( \sqrt{6ax}\frac{x\sqrt{x}}{32} \right )^{2a}_{0} +\left ( \frac{x}{2} \sqrt{\left ( 4a \right )^{2}-x^{2}}+\frac{16a^{2}}{2}\sin^{-1}\frac{x}{4a}\right )^{4a}_{2a}\right ]](https://entrancecorner.codecogs.com/gif.latex?%3D%202%5Cleft%20%5B%20%5Cleft%20%28%20%5Csqrt%7B6ax%7D%5Cfrac%7Bx%5Csqrt%7Bx%7D%7D%7B32%7D%20%5Cright%20%29%5E%7B2a%7D_%7B0%7D%20+%5Cleft%20%28%20%5Cfrac%7Bx%7D%7B2%7D%20%5Csqrt%7B%5Cleft%20%28%204a%20%5Cright%20%29%5E%7B2%7D-x%5E%7B2%7D%7D+%5Cfrac%7B16a%5E%7B2%7D%7D%7B2%7D%5Csin%5E%7B-1%7D%5Cfrac%7Bx%7D%7B4a%7D%5Cright%20%29%5E%7B4a%7D_%7B2a%7D%5Cright%20%5D)
![\Rightarrow \frac{4\sqrt{6a}}{3}\left ( 2a\sqrt{2a} \right )+\left [ 0+16a^{2}\times \frac{\pi }{2} \right ]-\left [ 4a^{2}\sqrt{3}+16a^{2}\times \frac{\pi }{6} \right ]](https://entrancecorner.codecogs.com/gif.latex?%5CRightarrow%20%5Cfrac%7B4%5Csqrt%7B6a%7D%7D%7B3%7D%5Cleft%20%28%202a%5Csqrt%7B2a%7D%20%5Cright%20%29+%5Cleft%20%5B%200+16a%5E%7B2%7D%5Ctimes%20%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%5Cright%20%5D-%5Cleft%20%5B%204a%5E%7B2%7D%5Csqrt%7B3%7D+16a%5E%7B2%7D%5Ctimes%20%5Cfrac%7B%5Cpi%20%7D%7B6%7D%20%5Cright%20%5D)
consider
solving (i) and (ii) ,
As
so req. area