The base of a solid in the xy-plane is the circle x2 + y2 = 16. cross sections of the solid perpendicular to the y-axis are equilateral triangles. what is the volume, in cubic units, of the solid?

Recall that the area of an equilateral triangle with side lengths is . If cross sections are to be taken perpendicular to the y-axis, then each section's side length will be determined by the horizontal distance between the right and left sides of the circle.

Since , you have , where the positive root corresponds to the right half and the negative roots corresponds to the left half.

The volume is given by




where the last equality follows from the fact that the integrand is symmetric about . The volume is then

It has a slope of 1 nd a y intercept of -2