Math 230 Sections 6,2Final Review Problems4/21/06Topics to study for final exam from first part of course.1. Vector addition, scalar product, dot product, cross product.2. Simple properties of space curves3. Distance between geometric objects in space4. Properties of gradient5. Chain rule for functions of several variables6. Double and triple integrals0.Find the area of the portion of the plane 1 =z+y+xthat liesinside the cylinderr= 2.1.Find the surface area of the portion of the sphere of radius 4 to theright of the planey= 2.2.Find the surface area of the part of the coney=√x2+z2betweeny= 1 andy= 3.3.Find the mass of the in the first quadrant bounded by the planex+y+z= 1 if its density isf(x, y, z) = 1-x-y4.Find the mass of the solidEin the bounded by the paraboloidz= 1-x2-y2amd thexy-plane if its density is given byf(x, y, x) =1-x2-y25.Find the mass of the solidEbounded by the spherex2+y2+z2= 4and inside the cylinder of radius if its density is given byf(x, y, x) =√4-x2-x26.Find the mass of a sphere of radius 2 centered at the origin havingdensity at any point equal to the distance squared of that point to thez-axis. distance11.Find the mass of a sphere of radius 2 centered at the origin havingdensity at any point equal to the distance squared of that point to theorigin.12.Find the volume that lies above the coneφ=π/4 and below thesphereρ= 4 cosφ.13.Evaluate the following triple interated integral by converting it tospherical coordinatesI=Z√2-√2Z√2-y2-√2-y2Z√4-x2-y2√x2+y2px2+y2dz dx dy14.Find the integralRCfdswheref(x, y, z) =x-y+zandCis theline segment joining (0,0,0) and (1,2,3).15.Find the integralRCfdswheref(x, y, z) is the same as in theprevious problem andCis the union of two line segments, one joining(0,0,0) to (1,0,3) and the second joining (1,0,3) to (1,2,3).16.Find the mass of a thin wire in thexy-plane consisting of a linefrom (1,0) to (0,1) and another line from (-1,0) whose density is|x|.17.Find the work done by a forceF=< y-x, z-y, x-z >alongthe curveC:r(t) =< t, t2, t3>,0≤t≤1.18.Find the work done by a forceF=< y-x, z-y, x-z >alongthe curve-Cwhich is the same curve as in Prob 1 but traversed in thereverse direction.21.IfF(x, y, z) =<3x2y2+ex,2x3y+cosy,cosz >then find a poten-tial function onR3for it. EvaluateRCF·drwhereC:r(t) =< t3, t5, t7>with 0≤t≤1.