1. (M) Let R be the region enclosed by the ellipse + 4y2 = 4. Using...

 

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1. (M) Let R be the region enclosed by the ellipse + 4y2 = 4. Using an appropriate change of variables to evaluate ffy2 dA. Make sure you specify the change of variables and draw the new region. 2. (M) Find the volume V of the solid region D bounded above by z = 5 on the sides by the paraboloid 2 = 16-1 - y and below by the ry-plane. 3. (M) Evaluate fF.dr where F = xyi + 2zj + (y+z)k where C is composed of the line segments from (0,0,0) to (0,-1,0) and from (0, -1,0) to (1, 1, 2) 4. (W) Evaluate Sc(e + y) dr + (x+2y) dy where C is any piecewise smooth curve in the ry-plane from (0, 1) to (2, 3). 5. (W) Use Green's theorem to evaluate fey + y dr + 3yr dy where C is the circle r + y = 100 oriented counterclockwise. 6. (W) Let be the semi-ellipsoid z = 41 - r - y oriented so that the normal is directed upward and let F = ri+ yj + z tan(ry)k. Compute ff curl(F).ndS (Hint: Use Stokes' theorem twice so that you express the required integral as the integral over a different region over which the quantities are easier to integrate). 1. (M) Let R be the region enclosed by the ellipse + 4y2 = 4. Using an appropriate change of variables to evaluate ffy2 dA. Make sure you specify the change of variables and draw the new region. 2. (M) Find the volume V of the solid region D bounded above by z = 5 on the sides by the paraboloid 2 = 16-1 - y and below by the ry-plane. 3. (M) Evaluate fF.dr where F = xyi + 2zj + (y+z)k where C is composed of the line segments from (0,0,0) to (0,-1,0) and from (0, -1,0) to (1, 1, 2) 4. (W) Evaluate Sc(e + y) dr + (x+2y) dy where C is any piecewise smooth curve in the ry-plane from (0, 1) to (2, 3). 5. (W) Use Green's theorem to evaluate fey + y dr + 3yr dy where C is the circle r + y = 100 oriented counterclockwise. 6. (W) Let be the semi-ellipsoid z = 41 - r - y oriented so that the normal is directed upward and let F = ri+ yj + z tan(ry)k. Compute ff curl(F).ndS (Hint: Use Stokes' theorem twice so that you express the required integral as the integral over a different region over which the quantities are easier to integrate).

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1 To evaluate the integral SRY dA over the region R enclosed by the ellipse 2 4y 4 we can make the f... View the full answer