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# Calculus II MATH 116

UM

GPA 3.87

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This 1 page Class Notes was uploaded by Mrs. Preston Lehner on Thursday October 29, 2015. The Class Notes belongs to MATH 116 at University of Michigan taught by Staff in Fall. Since its upload, it has received 7 views. For similar materials see /class/231509/math-116-university-of-michigan in Mathematics (M) at University of Michigan.

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Date Created: 10/29/15

Some Remaining Review Problems7Solutions This is mam camp rehensive n07 guaranteed m 272 useful 1 Find the center of mass T g of the solid formed by the region bounded by z 0 z 2 y l 6 and y l 7 6 a V Rotated about the yaxis if its density is 6y y mass unitsunit volumei Soluti0n7Draw a gure to see the shape Rotated about the y axis with a density 6y that only depends on y we can see by symmetry that E must be zero To nd y we need to nd the mass rst A volume slice at a height y with a thickness Ay is V5 7rlnl 7 y2 Ay if 0 S y S 17 6 2 V51 7r22Ay if 1 7 6 2 S y S 16 2 and V5 7rlny 7 12 Ay if 1 6 2 S y S 2 So the mass is 2 1762 1e 2 2 6y V5 y7rlnl7y2dy y7r22dy y7rlny7l2dy 0 0 1472 1e72 47r17 3e Then E is the moment over the mass where the moment is found by repeating the mass calculation with an additional factor of y 2 moment y 6y V5 0 1762 152 2 y2 7rlnl 7 y2 dy y2 7r22 dy y2 7rlny 712 dy 0 1752 1e 2 112 28 5 2 27 7r 7 we 712we Taking the moment over the mass we get g W 10612 Rotated about the zaxis if its density is 61 1 mass unitsunit volumei Soluti0n7similarly draw a gure to see the shape This is easier because we don7t have to break the region up to do the integral in this case we get 2 mass 17rl e E2 717 6 1 dz 47rl 7 3672 0 2 moment 12 7r 1 eix2 717 eix2 dz 87rl 7 5672 0 and so E 1i08864i 2 if for the region in 1a we had 61 3 I can we nd T Q Why or why not Soluti0n7We canlt calculate either of these easily with the tools that we have at our disposal in Math 116 You can see this by thinking about how we have to slice the region The fact that density is a function of 1 means that we want to slice the region so that slices are perpendicular to the zaxis This is ne but the solid doesn7t have any nice shape in that direction so it becomes very difficult to gure out the mass in this case Of course if the density only depends on I and the object is symmetric with respect to y y l in this case then we can deduce E from a symmetry argument y 1 But if we were to try to nd the moment needed to calculate y it will require us to have an integral with a factor of y in it which would mean that we have to slice the region perpendicular to the yaxis7which doesn7t work at all because we already know we have to slice it with respect to I because of the density Thus for T we7re stuck because it s very difficult to gure out what the slice of volume looks like to nd the mass and if we try to nd E using calculus the problem is further complicated because we canlt logically set up an integral for the moment that we need to nd the center of mass

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