suppose that two cylinders each with radius 1 intersect at right angles and their centers also intersect. What is the volume of the intersect?

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We have two cylinders, each with radius 1, intersecting at right angles. The first cylinder has its axis along the z-direction, described by x squared plus y squared less than or equal to 1. The second cylinder has its axis along the x-direction, described by y squared plus z squared less than or equal to 1. We need to find the volume of their intersection. The intersection is defined by both inequalities: x squared plus y squared less than or equal to 1, AND y squared plus z squared less than or equal to 1. We'll solve this using the method of slicing. We consider cross-sections perpendicular to the y-axis. For any fixed value of y between negative 1 and 1, we can determine the bounds for x and z coordinates. For a fixed value of y between negative 1 and 1, both inequalities give us bounds on x and z coordinates. From the first cylinder, x is bounded by plus or minus square root of 1 minus y squared. From the second cylinder, z has the same bounds. This means the cross-section at height y is a square in the x-z plane with side length 2 times square root of 1 minus y squared. The area of each square cross-section is 2 square root of 1 minus y squared, all squared, which simplifies to 4 times 1 minus y squared. To find the total volume, we integrate this area function from y equals negative 1 to y equals 1. This gives us the integral of 4 times 1 minus y squared dy, which equals 4 times the integral of 1 minus y squared dy. Now let's evaluate the integral. 4 times the integral of 1 minus y squared dy from negative 1 to 1 equals 4 times y minus y cubed over 3, evaluated from negative 1 to 1. Substituting the limits: 4 times 1 minus 1 over 3, minus negative 1 minus negative 1 over 3. This simplifies to 4 times 2 thirds minus negative 2 thirds, which equals 4 times 2 thirds plus 2 thirds. That's 4 times 4 thirds, giving us 16 over 3. Therefore, the volume of the intersection is 16 thirds.