Answer:
The given surfaces are z = x² + y² and x² + y² = 1. Here, we have to find the region bounded by the surfaces z = x² + y² and x² + y² = 1 for 1 ≤ z ≤ 4.
We have to find the equations for the traces of the surfaces at z = 4, y = 0, x = 0, and where the two surfaces meet using z.
Using cylindrical coordinates, we have x = rcosθ and y = rsinθ.
Then the surfaces can be written as follows.r² = x² + y² ... (1)z = r² ... (2)At z = 4, using equation (2), we get r = 2.
At y = 0, using equations (1) and (2), we get x = ±1.At x = 0, using equations (1) and (2), we get y = ±1.
Using equations (1) and (2), we get z = 1 at r = 1. So, the equations for the traces of the surfaces are as follows.
The inner trace at z = 4 is x² + y² = 4.The outer trace at z = 4 is x² + y² = 1.
The inner trace at y = 0 is z = x².
The outer trace at y = 0 is z = 1.
The inner trace at x = 0 is z = y².
The outer trace at x = 0 is z = 1.
The equation for where the two surfaces meet using z is x² + y² = z.
Explanation:
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