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##### MATH 243 Final Exam, Winter 2014 Paper

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solution

**Question**

Question;1. (a) Find the area of the parallelogram;with verticesA(????2,1),B(0,4),C(4,2),andD(2,????1);(b) Find the volume of the parallelepiped;determined by the vectorsa=;b;= andc=. [10 points];2. Find the area of the region inside the;circle (x????1)2+y2= 1 and outside the circle;x2+y2= 1.;[10;points];3. Determine whetherL1:x=y=z, L2:x+ 1 =y/2 =z/3 are parallel, skew, or;intersecting. If;skew;nd the distance between them. [12 points];4. Find the velocity and position vectors of;a particle that has the given acceleration and the given initial velocity and;positiona(t) = 2i+ 6tj+ 12t2k,v(0) =i,r(0) =j????k. [10 points];5. Evaluate the line integralZCF_dr, whereF= andCconsists of the line;segments from (0,0);to (2,1) and from (2,1) to (3,0). [12 points];6. A solidElies within the cylinderx2+y2= 1, below the planez= 4 and above the;paraboloid;z= 1????x2????y2. The density function is_(x, y) =px2+y2. Find the mass ofE. [10 points];7. Find the absolute maximum and minimum;values off;=x2+y2????2xon the setD, whereDis;the closed triangular region with vertices (2,0), (0,2), and (0,????2). [12 points];8. Evaluate the integral;Z1;0;Zp;1????x2;0;Zp;2????x2????y2;p;x2+y2;xydzdydx;[12 points];9. Evaluate the surface integralZZSF_dS, whereF= andSis the surface of the solid;bounded by the cylindery2+z2= 1 and the planesx=????1 andx= 2. [12 points].;10. (a) Show that the equation of the tangent;plane to the ellipsoidx2/a2+y2/b2+z2/c2=R2atP(x0, y0, z0) can be written asxx0/a2+yy0/b2+zz0/c2=R2;(b) Use the result of part (a) to _nd the;tangent plane tox2/4 +y2/4 +z2= 1 at the point (????1,1;P2;2;[Extra 10 points]

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