Question

Find the area of the ellipse x^2/25+y^2/16=1

Answer 1

x²/(5)² + y²/(4)² = 1

take y = 0

x²/(5)²= 1
x² = 5²
x = ± 5
e.g ellipse cut the x -axis at two points .( 5 , 0 ) and ( -5 , 0)

now,
y² /(4)² = 1 -x²/(5)²

y² = 4²( 5² -x²)/5²

y = ± 4/5√( 25 -x²)

it means ellipse just like sum of two functions which gives the area of ellipse .
so, area of ellipse = 2× area of function by 4/5√( 25 -x²)

now ,

area enclosed by y = 4/5√(25-x²) use integration

A = ydx = 4/5√(5²-x²)dx
= 4/5{ x/2√(25-x²) +25/2sin-¹x/5}
now put limit x = -5 to x = 5

A = 4/5{ 0 + (25/2)(π/2) +(25/2)(π/2)}
=4/5(25π/4 + 25π/4 )
=(4/5)(50π/4)
=10π

so, area of ellipse = 2× A = 2× 10π = 20π square units