Final answer:
To find the volume common to the two cylinders, set their equations equal to each other and determine the intersection. Find the cross-sectional area and height of the intersection, then integrate to find the common volume.
Step-by-step explanation:
To find the volume common to the two cylinders x²+y²=4 and x²+z²=4, we need to determine the intersection between the two cylinders. By setting the equations equal to each other, we get y² = z². This means that the heights of the cylinders are equal.
The radius of each cylinder is given by r = √(4-x²). To find the volume of the intersection, we integrate the cross-sectional area of the common region over the range of x where both cylinders exist. The cross-sectional area is given by A = πr², and the height of the intersection is h = √(4-x²).
Integrating A*h over the range where the two cylinders intersect will give us the volume common to the two cylinders.