Problems

1            The base of a solid is the triangle in the x, y-plane with vertices at (0, 0), (0, 1), and (1, 0). The cross sections perpendicular to the x-axis are squares with one side on the base. Find the volume of the solid.

2            The base of a solid is the region in the x, y-plane bounded by the parabola y = x² and the line y = 1. The cross sections perpendicular to the x-axis are squares with one side on the base. Find the volume of the solid.

3            Find the volume of the solid in Problem 1 if the cross sections are equilateral triangles with one side on the base.

4            Find the volume of the solid in Problem 2 if the cross sections are equilateral triangles with one side on the base.

5            Find the volume of the solid in Problem 1 if the cross sections are semicircles with diameter on the base.

6            Find the volume of the solid in Problem 2 if the cross sections are semicircles with diameter on the base.

7            Find the volume of a wedge cut from a circular cylinder of radius r by two planes whose line of intersection passes through the axis of the cylinder, if the wedge has thickness c at its thickest point.

8            Find the volume of the smaller wedge cut from a circular cylinder of radius r by two planes whose line of intersection is a chord at distance b from the axis of the cylinder, if the greatest thickness is c.