Final answer:
The base of the solid is a circle and the cross-sections perpendicular to the y-axis are equilateral triangles. To find the volume of the solid, we need to integrate the cross-sectional area over the range of y-values.
Step-by-step explanation:
The base of the solid is the circle x²+y² = 4. The cross-sections perpendicular to the y-axis are equilateral triangles. To find the volume of the solid, we need to integrate the cross-sectional area over the range of y-values. Since the cross-sections are equilateral triangles, the area can be calculated as A = (sqrt(3)/4) * s^2, where s is the side length of the triangle.
For a given y-value, the side length of the equilateral triangle can be determined using the equation of the circle: x² + y² = 4. Solving for x, we get x = sqrt(4 - y²). Substituting this into the equation for the area, we have A = (sqrt(3)/4) * (sqrt(4 - y²))^2 = sqrt(3 - 3y²/4).
To find the volume (V), we integrate the area over the range of y-values from -2 to 2 (the range of the given circle). Therefore, V = (sqrt(3)/4) * integral from -2 to 2 of sqrt(3 - 3y²/4) dy.