Test4ReviewSolutions - Test 4 Review 1 Find the length of...

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Test 4 Review 1. Find the length of the curve r ( t ) = (cos 3 t ) i + (sin 3 t ) k , 0 t π/ 2. Solution: 3 / 2 2. A wire is given by the straight-line segment connecting (0 , 1 , 0) to (1 , 0 , 0) and has density δ = x + y . Find the mass of the wire. Solution: 2 3. Calculate the work done by the force F = 3 x 2 - 3 x, 3 z, 1 from (0 , 0 , 0) to (1 , 1 , 1) over the path r ( t ) = t, t 2 , t 4 , 0 t 1. Solution: 3 / 2 4. Let F = sin y cos x i +cos y sin x j + k . Show that F is conservative and find a potential function. Calculate the integral C sin y cos xdx + cos y sin xdy + dz where C is the straight-line from (1 , 0 , 0) to (0 , 1 , 1). Solution: F = (sin x sin y + z + C ); 1 5. Use Green’s Theorem to evaluate (6 y + x ) dx + ( y + 2 x ) dy where C is the (counter-clockwise) circle ( x - 2) 2 + ( y - 3) 2 = 4. Solution: - 16 π 6. Find the area of the surface x 2 - 2 y - 2 z = 0 that lies above the triangle bounded by the lines x = 2 , y = 0 , y = 3 x in the xy -plane. Solution: 6 6 - 2 2 7. Integrate g ( x, y, z ) = x + y + z over the portion of the plane 2 x + 2 y + z = 2 that lies in the first octant.
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Solution: 2 8. Find the flux of F = x, y, z
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