Volume of Solid w/ Elliptical Base & Right Triangles

[vigintitres]

Homework Statement

The base of S is an elliptical region with boundary curve 9x^2 + 4y^2 = 36. Cross-sections perpindicular to the x-axis are isosceles right triangles with hypotenuse in the base.

Find the volume of the described solid.

Homework Equations

V = {int} 1/2 b*h dy

The Attempt at a Solution

I found that I would have to use the symmetry to solve this. The only things I have are x^2 + y^2 = 1/2 and y = sqrt(.5 - x^2)

Now i know I have to integrate an isosceles triangles area which is 1/2 b*h but I'm not sure what the base or the height will be.

 

[tiny-tim]

Hi vigintitres!

The base of S is an elliptical region with boundary curve 9x^2 + 4y^2 = 36. Cross-sections perpindicular to the x-axis are isosceles right triangles with hypotenuse in the base.

Find the volume of the described solid.
…
I found that I would have to use the symmetry to solve this. The only things I have are x^2 + y^2 = 1/2 and y = sqrt(.5 - x^2)

Where do you get x^2 + y^2 = 1/2 from?

9x^2 + 4y^2 = 36.

Now i know I have to integrate an isosceles triangles area which is 1/2 b*h but I'm not sure what the base or the height will be.

The base is 2y.

The height you can work out because it's a right-angled isoceles triangle.

 

[ZharAngel]

Find volume via method of cross-sections.