MAT2500-03/04 10S Final Exam Print Name (Last, First) _________________________________|__Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use arrows and equal signs when appropriate. Always simplify expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation). Indicate where technology is used and what type (Maple, GC). Use technology to check any integrals you set up.1. Given the point x, y, z =K3, 4, 12 , find the new coordinates, in each case stating the angles both in radians (exactly, using inverse trig functions) and in degrees (1 decimal place accuracy) and use proper identifying symbols for all coordinates: a) cylindrical coordinates. b) spherical coordinates. Support your work with two diagrams, one of the xy plane and one of the rz half plane, each including a reference triangle locating the point with respect to the axes with all three sides labeled by their lengths and both axes labeled by their coordinate labels and showing the relevant angles. Show clearly how you obtain values of your coordinates from these diagrams. Do the angles look right in these diagrams?0 0.5 1 1.500.511.52Ry = xx2Cy2=4yx2. a) Describe the region R by giving the appropriate intervals of the polar coordinates over the region, and draw in the diagram a typical radial cross-section, labeling its endpoints by the values of the radial coordinate, shading in the region with equally spaced radial cross-sections. State the ranges of the two polar coordinates as inequalities.b) Use polar coordinates to evaluate the three integrals A =R1 dA, Ax=Rx dA and Ay=Ry dA by hand exactly. Evaluate the coordinates x = Ax/A, y = Ay/A of the centroidof the region R exactly and numerically. Locate the centroid on the diagram. Does it seem right? Explain. 3. a) Check that F=2 xK2 y,K2 xC2 y satisfies the condition that it admit a potential function, i.e., is a conservative vector field.b) Find a potential function f for it.c) Use the potential to evaluate the line integral CF ,d r over any curve from 2, 0 to 0, 3 .d) By Green's theorem CF ,d r =R vF2v xKvF1v y dA , the line integral around the triangle from 0, 0 to 2, 0 to 0, 3 to 0, 0 is zero. Check this by evaluating directly (not using the potential) the line integral over the three sides of this triangle with the counterclockwise direction, i.e., show that their sum is zero. Make a diagram showing your attack of the problem. [Note that your line integral over the hypotenuse should equal part c) as a check. You can also separately check the other two line integrals in this way.]solution (on-line)pledgeWhen you have completed the exam, please read and sign the dr bob integrity pledge if it applies and hand in with your answer sheets as a cover page, with the Lastname, FirstName side face up: "During this examination, all work has been my own. I give my word that I have not resorted to any ethically questionable means of improving my grade or anyone else's on this examination and that I have not discussed thisexam with anyone other than my instructor, nor will I until after the exam period is terminated for all participants."Signature: