Need Detailed Explanation of Bonus 3,4, D,F,E and there parts on Paper

D Bonus 3 Choose one of the below. An extra page has been left for workspace if you feel you need to use it. 40 pts 1) Let F = (z + y.z, y + x). Determine whether or not F is conservative. 15 pts Find the volume of the region in space bounded by the below equations. You may use cither a double or a triple integral. 2=0 z=/(x,y) = erty x=0 ) x = Iny y=1 y = In 8) 2) Find the area of the region in the plane bounded by the below equations. You must use a double integral. (y=0 y=8 y=x] 3) Find the volume of the solid region bounded above by the paraboloid given by z= /(x,y) = 9- x - y' and below by the unit circle in the xy plane. You may use either a double or a triple integral. Hint: a change of coordinates would be helpful. 2) Evaluate the line integral below for your choice of one of the paths in a,b. 25 pts (2x - 3y + 1) dx - (3x + y -5) dy Scanned with CamScanner a. C: Semicircular path from (0, -1) to (1, 0) to (0, -1). b. C: Path from (0, 1) to (2, e ) along y = ex. E Choose one of the below. An extra page has been left for workspace if you feel you need to use Scanned with CamScanner 50 pts 1) Use a transformation (involving a Jacobian) to evaluate the below integral. Hint: the stuff under the radicals or in parentheses would be a great place to look for T-1. ['S Sixty (-2x) dy dx Bonus 4 2) Use cylindrical coordinates to evaluate the below. Work your choice of one of the following: 40 pts Where - S z S and the "shadow" of S in the xy plane is the unit circle. I) Find an equation of the tangent plane at (0, 6,4) to the parametric surface given by r(u, v) = (2ucosv, 3usinv, u") 3) Use spherical coordinates to find the volume of the solid region above the xy plane and between the sphere p = cos of and the hemisphere p = 2. 2) Let C be the triangular region in QI of the xy plane bounded by (x =1 y= 0 y= x]. Let F = (2xy , 4x y? ). Find F . dr Scanned with CamScanner 3) Let C be the triangular region in QI of the xy plane bounded by (x=0 y=0 y= 1- x). Find yz dx + x? dy Choose one of the below. An extra page has been left for workspace if you feel you need to use 20 pts 1) Reverse the order of integration of the double integral and evaluate. integral. 2) Convert the below integral to cylindrical coordinates. You do not need to evaluate the say dz dy dx 3) Convert the below integral to spherical coordinates. Let S be the region bounded above by a hemisphere of radius 4 and below by the xy plane. You do not need to evaluate the 4) Set up (but do not evaluate) your choice of a double or triple integral to find the volume of the region in the first octant beneath the plane given by 3x + y + 2z = 6