The area of the region bounded by the ellipse `(x^(2))/25+y^(2)/16=1` is

Options

A. `20pi" sq units"`
B. `20pi^(2)" sq units"`
C. `16pi^(2)" sq units"`
D. `25 pi " sq units"`

Answer

A

Explanation

We have, `(x^(2))/5^(2)+y^(2)/4^(2)=1`
Here, a=pm5" and "b=pm4`
and y^(2)/4^(2)=1-x^(2)/5^(2)`
`rArr y^(2)=16(1-x^(2)/25)`

`rArr y=sqrt(16/25(15-x^(2))`
`rArr y=4/5sqrt(5^(2)-x^(2))`
`:. "Area enclosed by ellipse,"A=2. 4/5int_(-5)^(5)sqrt(5^(2)-x^(2))dx`
`=2. 8/5int_(0)^(5)sqrt(5^(2)-x^(2))dx`
`=2. 8/5[x/2sqrt(5^(2)-x^(2))+(5^(2))/2sin^(-1).x/5]_(0)^(5)`
`=2. 8/5[5/2sqrt(5^(2)-5^(2))+5^(2)/2sin^(-1). 5/5-0-25/2 .0]`
`=2. 8/5[25/2 .pi/2]`
`=16/5 . (25pi)/4`
`=20pi" sq units"`