Let C = C₁ UC2, where C₁ is the semicircle x = √√4y - y² traced from (0, 4) to (0, 0) while C₂ is the line segment from (0, 0) to (0,4). a. Use a line integral to find the area of the surface S := {(x, y, z) € R³ : (x, y) € C₁, 0 ≤ z ≤ x²}. b. Use Green's Theorem to evaluate J₁ xy dx - x² dy.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Let C = C₁ UC2, where C₁ the semicircle x = √4y - y² traced from (0,4) to (0,0) while
C₂ is the line segment from (0,0) to (0,4).
a. Use a line integral to find the area of the surface
S := {(x, y, z) € R³ : (x, y) = C₁, 0 ≤ z ≤ x²}.
b. Use Green's Theorem to evaluate
[xy dx = x² dy.
Transcribed Image Text:Let C = C₁ UC2, where C₁ the semicircle x = √4y - y² traced from (0,4) to (0,0) while C₂ is the line segment from (0,0) to (0,4). a. Use a line integral to find the area of the surface S := {(x, y, z) € R³ : (x, y) = C₁, 0 ≤ z ≤ x²}. b. Use Green's Theorem to evaluate [xy dx = x² dy.
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