can anyone help me w/ this volume??

Question

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

Volume = ?

Answer 1

Rewrite the boundary as:
y = +- 3*sqrt(4 - x^2)/2
Since the volume is symmetrical to both y-z and x-z planes, we only need to find the volume in the first quadrant, and time it by 4. For a thin slice with thickness of dx, the shape is a isosceles right triangle with leg y, thus the top area of this slice is y^2/2 = 9(4 - x^2)/8. The volume of this slice is 9(4 - x^2)dx/8. integrate this from 0 to 2 yields (*4 to get final result):
9/2 * Int (4-x^2) =9/2 * (8-8/3) = 24