Question

As viewed from above, a swimming pool has the shape of the ellipse
(x^2)/4900+(y^2)/2500=1,
where x and y are measured in feet. The cross sections perpendicular to the x-axis are squares. Find the total volume of the pool.

Answer 1

so, each slice of the volume has base 2y, making its area 4y^2
From the equation, 4y^2 = 4*2500(1-x^2/4900)
so, the volume of the pool is the sum of the volumes of all those squares of thickness dx. Using symmetry, that means
v = 2∫[0,70] 10000(1 - x^2/4900) dx

Answer 2

The equation of the ellipse:

x² / a² + y² / b² = 1

x² / 4900 + y² / 2500 = 1

a² = 4900

a = √4900 = 70

b² = 2500

b = √2500 = 50

Because we have the origin of our coordinate system in the center of the pool, the ycoordinate for each value of x equals half of the width of the pool at that location.

This means that the dimensions of each square-shaped slice are:

width = 2 y

depth = 2 y

where y is a function of x

The thickness of each slice is equal to dx

The area of each cross-sectional slice is:

( 2 y ) ∙ ( 2 y ) = 4 y²

The volume of each cross-section is:

dV = 4 y² dx


x² / a² + y² / b² = 1

Rearranging the equation gives:

y² = b² ∙ ( 1 - x² / a² )


This can be substituted into the equation for dV:

dV = 4 y² dx = 4 ∙ b² ∙ ( 1 - x² / a² ) ∙ dx

Due to symmetry, you can calculate the volume of half of the pool, and then multiply by 2 to get the total volume.

V = 2 ∫ dv

V = 2 ∫ 0 to a) ∙ 4 ∙ b² ∙ ( 1 - x² / a² ) ∙ dx

V = 2 ∙ 4 ∙ b² ∫ ( 0 to a) ∙ ( 1 - x² / a² ) ∙ dx

V = 8 b² ∫ ( 0 to a) ( 1 - x² / a² ) ∙ dx

________________________________
Remark:

∫ ( 1 - x² / a² ) = x - x³ / 3 a² + C
_________________________________

V = 8 b² [ x - x³ / 3 a² ] 0 to a


8 b² [ x - x³ / 3 a² ] of x = a is:

8 b² ∙ ( a - a³ / 3 a² ) = 8 b² ∙ ( a - a / 3 ) =

8 b² ∙ ( 3 a / 3 - a / 3 ) = 8 b² ∙ 2 a / 3 = 16 b² ∙ a / 3 = 16 ∙ a ∙ b² / 3


8 b² [ x - x³ / 3 a² ] of x = 0 is:

8 b² ∙ ( 0 - 0³ / 3 a² ) = 8 b² ∙ ( 0 - 0 / 3 a² ) =

8 b² ∙ ( 0 - 0 ) = 8 b² ∙ 0 = 0


V = 8 b² [ x - x³ / 3 a² ] 0 to a = 16 ∙ a ∙ b² / 3 - 0

V = 16 ∙ a ∙ b² / 3

Substituting a = 70 and b = 50 gives:

V = 16 ∙ 70 ∙ 50² / 3

V = 16 ∙ 70 ∙ 2 500 / 3

V = 16 ∙ 175 000‬ / 3

V = 2 800 000‬ / 3

V = 933333.333... ft³