Sketch the region R= {(x, y) ER: x y x, 1 < x1}, and...

 

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Sketch the region R= {(x, y) ER: x² ≤ y ≤ x², −1 < x≤1}, and evaluate the double integral (b) Calculate the work integral SS₁zªda. [10] E-dr. where F-yi-ri-3r³y²z k and C is the counter-clockwise semi-circle in the ay-plane (i.e. ==0) starting at (3,0,0) and ending at (-3,0,0). [10] (c) State carefully Stoke's theorem and use it to evaluate G-dr. where Gy³i+2³zj-a²yk and C is the (counterclockwise) circle r²+²=4 parallel to the ry-plane at = 1. [10] Sketch the region R= {(x, y) ER: x² ≤ y ≤ x², −1 < x≤1}, and evaluate the double integral (b) Calculate the work integral SS₁zªda. [10] E-dr. where F-yi-ri-3r³y²z k and C is the counter-clockwise semi-circle in the ay-plane (i.e. ==0) starting at (3,0,0) and ending at (-3,0,0). [10] (c) State carefully Stoke's theorem and use it to evaluate G-dr. where Gy³i+2³zj-a²yk and C is the (counterclockwise) circle r²+²=4 parallel to the ry-plane at = 1. [10]