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Unformatted text preview: Math 230 Sections 6,2 Final Review Problems 4/21/06 Topics to study for final exam from first part of course. 1. Vector addition, scalar product, dot product, cross product. 2. Simple properties of space curves 3. Distance between geometric objects in space 4. Properties of gradient 5. Chain rule for functions of several variables 6. Double and triple integrals 0. Find the area of the portion of the plane 1 = z + y + x that lies inside the cylinder r = 2. 1. Find the surface area of the portion of the sphere of radius 4 to the right of the plane y = 2. 2. Find the surface area of the part of the cone y = x 2 + z 2 between y = 1 and y = 3. 3. Find the mass of the in the first quadrant bounded by the plane x + y + z = 1 if its density is f ( x, y, z ) = 1- x- y 4. Find the mass of the solid E in the bounded by the paraboloid z = 1- x 2- y 2 amd the xy-plane if its density is given by f ( x, y, x ) = 1- x 2- y 2 5. Find the mass of the solid E bounded by the sphere x 2 + y 2 + z 2 = 4 and inside the cylinder of radius if its density is given by f ( x, y, x ) = 4- x 2- x 2 6. Find the mass of a sphere of radius 2 centered at the origin having density at any point equal to the distance squared of that point to the z-axis. distance 11. Find the mass of a sphere of radius 2 centered at the origin having density at any point equal to the distance squared of that point to the origin. 12. Find the volume that lies above the cone = / 4 and below the sphere = 4 cos . 13. Evaluate the following triple interated integral by converting it to spherical coordinates I = Z 2- 2 Z 2- y 2- 2- y 2 Z 4- x 2- y 2 x 2 + y 2 p x 2 + y 2 dz dx dy 14. Find the integral R C fds where f ( x, y, z ) = x- y + z and C is the line segment joining (0 , , 0) and (1 , 2 , 3). 15. Find the integral R C fds where f ( x, y, z ) is the same as in the previous problem and C is the union of two line segments, one joining (0 , , 0) to (1 , , 3) and the second joining (1 , , 3) to (1 , 2 , 3). 16. Find the mass of a thin wire in the xy-plane consisting of a line from (1,0) to (0,1) and another line from (-1,0) whose density is | x | . 17. Find the work done by a force F = < y- x, z- y, x- z > along the curve C : r ( t ) = < t, t 2 , t 3 >, t 1. 18. Find the work done by a force F = < y- x, z- y, x- z > along the curve- C which is the same curve as in Prob 1 but traversed in the reverse direction. 21. If F ( x, y, z ) = < 3 x 2 y 2 + e x , 2 x 3 y + cos y, cos z > then find a poten- tial function on R 3 for it. Evaluate R C F d r where C : r ( t ) = < t 3 , t 5 , t 7 > with 0 t 1. 22. Consider the vectorfield G = 1 x 2 + y 2 < y,- x, > . Calculate R C F d r where is a clockwise oriented circle with center at some point on the z-axis. Why can you conclude that G is not conservative on the region D = R 3 less the z-axis....
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