Final answer:
The volume of the solid within the cylinder x^2 + y^2 = 1 and the sphere x^2 + y^2 + z^2 = 4 is found using triple integration in cylindrical coordinates, considering the geometry of the overlap region of the cylinder and sphere.
Step-by-step explanation:
To find the volume of the solid that lies within both the cylinder x2 + y2 = 1 and the sphere x2 + y2 + z2 = 4, we need to consider the geometric constraints imposed by these equations. The cylinder has a radius of 1 unit and is infinite in the z-direction, whereas the sphere has a radius of 2 units.
The overlap region between the sphere and the cylinder will be symmetrical about the z-axis, and we can calculate its volume by integrating the area of the circular cross-section of the cylinder along the z-axis, limited by the sphere.
First, take the equation of the sphere and solve for z to find the limits of integration:
- z2 = 4 - (x2 + y2)
- So z = ±√(4 - (x2 + y2)) which represents the upper and lower halves of the sphere.
Using cylindrical coordinates, the volume V is given by the integral:
V = ∫-11 ∫02π ∫ -√(4 - r2)√(4 - r2) r dzdθdr
By evaluating this integral, we can find the volume of the solid that lies within the given constraints.