MATH 244, Fall ’15 Final

1 math 244, fall 15 FinalName:INSTRUCTIONS:Write legibly. Indicate your answer clearly. Show all work;explain your answers. Answers with work not shown might be calculators, cell phones, or (15)Find the volume of the solid bounded below by the surfacez=x2+y2and above by the planez= shaddow of the solid is a disk of radius 2. Using polar coordinates we findVol = 2 0 20 4r2rdz drd = 2 0 20(4 r2)rdrd = 8 2.(10)LetDbe the solid in the first octant, below the plane 3x+ 6y+ 2z= 6. Suppose its desity is = byIzthe second moment ofDabout thez axis.(a) Express the moment as an integral (other orders of integration lead to different expressions for thelimits of integration):Iz= 20 1 x20 3 3x2 3y0(x2+y2)dz dy dx(b) For 5 points of extra credit, you may calculateIz. It turns out to be 1 (20)Consider the upper hemisphere of a ball of radius 5,x2+y2+z2 25 andz 0. Suppose the densityis (x,y,z) = 2z. Find the massM, first momentMx,y, andz coordinatezof the center of 2 0 20 50(2 cos )( 2sin )d d d =625 2Mx,y= 2 0 20 50( cos )(2 cos )( 2sin )d d d =2500 3z=Mx,yM=834.

2 (20)For the vector fieldF(x,y,z) = (0,0,z), find the flux SF nd across the portionSof the spherex2+y2+z2=a2in the first octant in the direction away from the parametrize the surface using spherical coordinates:r( , ) = (acos sin ,asin sin ,acos ) with 0 2 and 0 /2,and find thatr r = ( , , a2sin cos ). This means that SF nd = 20 20(0,0,acos ) ( , ,a2sin cos )d d = 20 20a3cos2 sin d d = a365.(25)Calculate the integral: 2/30 2 2yy(x+ 2y)ey xdxdy(a) For the substitutionu=x+ 2yandv=x y, expressxandyin terms ofuandvand findJsothatdxdy=J find thatx=13(u+ 2v) andy=13(u v) and we calculatedxdy= det(xuyuxvyv) dudv= det(1/31/32/3 1/3) dudv=13dudv.(b) Sketch the region of integration in terms region is a triangle bounded by thex-axis (y= 0), the diagonal (x=y), and the linex= 2 2y(equivalentlyy= 1 x2).(c) Describe the region of integration in terms thaty= 0 u=v,x=y v= 0, andx= 2 2y u= 2.

3 As region ofintegration we find a triangle that is bounded by theu axis (v= 0), the diagonalu=v, and thelineu= 2.(d) Calculate the integral: 2/30 2 2yy(x+ 2y)ey xdxdy=13 20 u0ue vdv du=e 2+ (20)Find the work done by a forceF(x,y,z) = (6z,y2,12x) over a curveCthat is parametrized byr(t) =(cost,sint,t/6) for 0 t 2 in the direction of our standard formula we findW= CF T ds= 2 0(t,sin2t,12 cost) ( sint,cost,1/6)dt= 2 0 tsint+ sin2cost+ 2 costdt= 2 .The last two summands integrate to zero, and for the first one we use integration by (20)LetCbe the curve in the plane that bounds the regiono x and 0 y sinx. Find C3y dx+ 2xdyWe setM= 3yandN= 2x. ThenNx= 2 andMy= 3. We apply Green s Theorem and find C3y dx+ 2xdy= 0 sinx0Nx Mydy dx 0 sinx0 1dy dx= 0sinxdx= 28.(20)Find the outward flux of the vector fieldF(x,y,z) = (x2,xz,3z) across the sphere of radius 2, centeredat the the sphere andBfor the ball thatSbounds.

4 Note that the divergence ofFis F= 2x+ the divergence theorem we compute:Flux = SF nd = B2x+ 3dV= 3 Vol(B) = 32 .Page 2