Question

consider the solid s whose base is the triangular region with vertices (0,0) , (1,0) , and (0,1) . cross-sections perpendicular to the y -axis are equilateral triangles. volume of s =

Answer 1

To find the volume of the solid S, we need to calculate the area of each equilateral triangle cross-section perpendicular to the y-axis and then integrate it over the entire length of the solid. The volume of solid S is √(3)/4.

Step-by-step explanation:

To find the volume of the solid S, we need to calculate the area of each cross-section perpendicular to the y-axis and then integrate it over the entire length of the solid. The cross-sections are equilateral triangles, which have a formula for the area: A = (√(3)/4) * s², where s is the length of the side of the triangle.

In this case, the side of the equilateral triangle is given by the distance between the y-axis and the line segment connecting (0,0) and (0,1), which is 1. So the area of each cross-section is A = (√(3)/4) * 1² = √(3)/4.

To find the volume, we integrate this area over the interval from y = 0 to y = 1. The volume V is then given by V = ∫(0 to 1) A(y) dy. Since the area A(y) is constant, we can take it out of the integral, giving V = A * ∫(0 to 1) dy. Thus, the volume of solid S is V = (√(3)/4) * 1 * ∫(0 to 1) dy = (√(3)/4) * y|0 to 1 = (√(3)/4) * (1 - 0) = √(3)/4.