Volume by cross-section: ellipse and equilateral triangle cross sections?? 1. The problem statement, all variables and given/known data The base of a solid is the region bounded by the ellipse 4x^2+9y^2=36. Find the volume of the solid given that cross sections perpendicular to the x-axis are: a) equilateral triangles b) squares 2. Relevant equations 3. The attempt at a solution So I'm not really sure how ellipses work.. how can I sketch this ellipse? Beyond that.. I try to calculate the area of the triangle and then integrate in terms of y so the base is changing according to the ellipse curve. I write the ellipse as: y = +/-sqrt((-4/9)x^2 + 4) So the base of the triangle is 2(sqrt((-4/9)x^2 + 4)) And has that as the length on all side since it is equilateral. Then I try to find the height using Pythagoras and get h = +/-sqrt((-4/3)x^2 + 12) Then now I have the area of the triangle as (1/2)bh, which is = A = (1/2)(2(sqrt((-4/9)x^2 + 4)))(sqrt((-4/3)x^2 + 12)) Then I can integrate in terms of x.. does that look correct so far?