Calculus III MA 113
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This 2 page Class Notes was uploaded by Eldon Trantow on Monday October 19, 2015. The Class Notes belongs to MA 113 at Rose-Hulman Institute of Technology taught by Kurt Bryan in Fall. Since its upload, it has received 14 views. For similar materials see /class/225079/ma-113-rose-hulman-institute-of-technology in Mathematics (M) at Rose-Hulman Institute of Technology.
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Date Created: 10/19/15
Mess 0 Double Integrals Key 1 The reversed order is 32 6 2 zdzdy 0 x4Ty 2 The integral for the volume is 4 2 2952ydydz 0 0 3 A thin metal plate is bounded by 0 S x S 5 and 0 S y S x25 The surface density of the plate is 0Ly 2 z y2 kg per square meter 5 125 dy dz 0 0 5 5 dx dy 0 V51 5 125 2xy2dydz 0 0 a Area is just Or in the other order7 b Mass is either of Or in the other order7 5 5 2xy2dzdy 0 V571 c A little dA piece of the plate has mass 039 dA7 so weight 90 dA When the plate is tipped upward this piece at position is lifted a distance ofy meters7 so the work required to lift this little piece is dW yga dA Total work required is 5 125 2 00 9y2zydydz if you do it in the order dA dy dx 4 well integrate in the order dy dx The area is 2 x47m2 dy dz 2 4m 2 14712 2 2 27 47 22y dydz Mass is 5 CT 5 Consider a small dA piece at position 27y at least that7s the position at some moment during the plates rotation The piece has mass UdA It lies at a distance r xxz y2 meters from the origin the pivot and so is moving at speed 1027Tr meters per second lt7 kinetic energy is U dA207rr2 just mvz This simpli es to kinetic energy 2007r22 xzyz 2 yz dy dx Total kinetic energy is the sum 2 2 4 m2 22 2 2 2007f 2 4 222y ydydx Volume is 1 xliz2 2i17x27y2dydx 71 7W This is tricky Ask me if you want a thorough explanation The integral is x 12712 WE 122 22dd Li x12712 7 z T y y m It turns out to be 7r4
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