practice final - Sample problems on Chapter 14 Problem 1...

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Sample problems on Chapter 14 Problem 1: Find the surface area of the torus T (shaped like a bagel) with surface described in parameterized form by x = (3 + cos t ) cos θ, y = (3 + cos t ) sin θ, z = sin t, where 0 t, θ 2 π . Problem 2: Evaluate the line integral of cos x + sin y on the curve C : y = x, 0 x π/ 2. Problem 3: Evaluate the integral of y over the surface S : z = cosh x, 0 x 1 , 0 y 1 . Problem 4: Find the work done by F = ( xy ) i + ( y ) j + ( yz ) k over the curve r ( t ) = t i + t 2 j + t k , 0 t 1. Problem 5: Let F = (2 y sin z ) j + ( y 2 cos z ) k represent a force field and consider a curved line C described by r ( t ) = t cos t 2 i + t sin t 2 j + t 2 k , 0 t p π/ 2. Calculate the work integral R C F · d r by any legitimate means. Problem 6: Given the vector field F = ( y + 3) i + ( x - z ) j + (sin z - y ) k and the path γ described by two connected straight line segments from (0 , 0 , 0) to (1 , 2 , 1) to (0 , 0 , 1) a) Find a parameterization for a straight line segment from (0 , 0 , 0) to (1 , 2 , 1). b) is the vector field F a gradient vector field? (You must support your an- swer).
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