Volume, Mass Of An Element (Triple Integral) 08 Ee 09
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The document discusses calculating the volume and mass of 3D objects using triple integrals. It provides examples of setting up triple integrals to calculate the volume of various prisms and solids bounded by planes and paraboloids, noting that the order of integrating variables is important and that the outer integral limits must be constants while inner limits can involve other variables. It also explains that mass can be calculated similarly to volume by introducing the density and integrating it over the volume.
Volume, Mass Of An Element (Triple Integral) 08 Ee 09
- 1. Volume, Mass of an Element (Using Triple Integral) Presented By: 08-EE-09
- 2. “ If the limits on the integrals involve some of the variables then the order in which the integrations are performed is crucial.”
- 3. Some notes on Triple Integrals: We used a double integral to integrate over a two-dimensional region and so it shouldn’t be too surprising that we’ll use a triple integral to integrate over a three dimensional region. Since triple integrals can be used to calculate volume, they can be used to calculate total mass (recall Mass = Volume * density) and center of mass When setting up a triple integral, note that The outside integral limits must be constants The middle integral limits can involve only one variable The inside integral limits can involve two integrals
- 5. If distance is in cm and k = 1 gram per cubic cm per cm, then the mass of the cube is 128 grams.
- 6. Example: Let D be the prism bounded by the 3 coordinate planes as well as the planes x = 2 and y + z = 1. Find the volume of D using triple integrations with all different orders of x, y and z .
- 7. Continued on next slide
- 9. Example: Let D be the solid bounded by the paraboloids z = 8 x 2 y 2 , and z = x 2 + 3 y 2 . Set up the limits of integrations for evaluating the triple integral: Solution:
- 11. Example Evaluate
- 12. Here is a sketch of the plane in the first octant.
- 14. Now we need the limits of integration. Since we are under the plane and in the first octant (so we’re above the plane z = 0 ) we have the following limits for z . We can integrate the double integral over D using either of the following two sets of inequalities .
- 16. Mass: The mass of an object can be formulated the same as its volume by introducing the density. dm = ρ dV Integrating over the distributed mass of the object, Assuming the density ρ remains constant through out the object we have, ∫ ∫ ∫ ρ dV m = m ∫ ∫ ∫ dV m = ρ = ρ V V