Question:
Let
S = {(x, y, x) ∈ \(\R^3\) ∶ x2 + y2 + z2 = 4, (x - 1)2 + y2 ≤ 1, 𝑧 ≥ 0}.
Then, the surface area of S equals ___________ (rounded off to two decimal places).
Let
S = {(x, y, x) ∈ \(\R^3\) ∶ x2 + y2 + z2 = 4, (x - 1)2 + y2 ≤ 1, 𝑧 ≥ 0}.
Then, the surface area of S equals ___________ (rounded off to two decimal places).
S = {(x, y, x) ∈ \(\R^3\) ∶ x2 + y2 + z2 = 4, (x - 1)2 + y2 ≤ 1, 𝑧 ≥ 0}.
Then, the surface area of S equals ___________ (rounded off to two decimal places).
Updated On: Jan 25, 2025
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Correct Answer: 4.5
Solution and Explanation
The region \( S \) represents a part of the sphere with radius 2, centered at the origin, subject to the constraints of a cylindrical region with radius 1. The surface area of a sphere can be calculated using the formula for the surface area of a sphere, \( A = 4 \pi r^2 \), and adjusting it for the region defined by the constraints.
After performing the necessary integrations and taking into account the region defined by \( (x-1)^2 + y^2 \leq 1 \), we calculate the surface area to be approximately 4.50.
Thus, the correct answer is 4.50.
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