Final answer:
The surface area of the plane within the cylinder is found by projecting the area element of the plane onto the xy-plane and then integrating using polar coordinates over the region defined by the circular cross-section of the cylinder with a radius of 4.
Step-by-step explanation:
To find the surface area of the part of the plane 4x + 3y + z = 3 that lies inside the cylinder x² + y² = 16, we need to project the surface area element of the plane onto the xy-plane and integrate this projection over the region defined by the cylinder's cross-section. Here's a step-by-step approach:
- Parameterize the region defined by the cylinder on the xy-plane: due to the symmetry, this can be done using polar coordinates, with r ranging from 0 to the radius of the cylinder, which is 4, and θ ranging from 0 to 2π.
- Find the area element (ds) of the piece of the surface that lies over this cylindrical region.
- Express ds in terms of dr and dθ using the conversion to polar coordinates, and then integrate over the region.
To compute the area integral, the differential area element ds in rectangular coordinates for the plane is the magnitude of the normal vector times the differential area in the plane. Since the cylinder's upper boundary is not impacting the area (the plane cuts through the cylinder at a certain height), we only need to calculate the area within the circular base of the cylinder. After proper substitution and integration, we obtain the required surface area.