E is the tetrahedron bounded by the planes x=0, y=0 z=0, x+y+z=1 ; ρ(x, y, z)=y

Name: Problem 42, Chapter 15: Calculus Early Transcendentals
Uploaded: 2020-10-30T02:56:56-08:00
Duration: 14 min 4 s
Channel: Chris Trentman
Description: E is the tetrahedron bounded by the planes x=0, y=0 z=0, x+y+z=1 ; ρ(x, y, z)=y

step 1 We're given a solid and a density function on this solid. We are asked to find the total mass of

E is the tetrahedron bounded by the planes x=0, y=0 z=0,...

Key Concepts

Setting Integration Limits

Determining the correct limits of integration is a critical step in solving triple integration problems. The integration limits are derived from the equations of the bounding surfaces, which in the case of a tetrahedron typically include coordinate planes and a plane like x+y+z=1. Properly establishing these limits ensures that the integration is carried out over the intended region.

Density Function in Volume Integration

In the context of applying triple integrals to physical problems, a density function describes how a quantity (typically mass) is distributed within a region. When the density function is given as a function of the coordinates, such as ?(x, y, z), it must be incorporated into the integral to accurately compute the total mass or other related properties.

Tetrahedron

A tetrahedron is a type of polyhedron composed of four triangular faces, three of which meet at each vertex. In integration problems, a tetrahedron often serves as the domain over which a function is integrated, and its boundaries are defined by plane equations. Understanding the geometric properties of a tetrahedron is essential for correctly setting up and evaluating integrals over its volume.

Triple Integrals

Triple integrals are used to compute quantities over a three-dimensional region, such as volume or mass. They involve integrating a function of three variables over a given domain. Mastering triple integrals requires the ability to set up the integral limits in an order that simplifies the computation and reflects the geometry of the integration region.

Transcript

00:01 We're given a solid and a density function on this solid.

00:05 We are asked to find the total mass of the solid and its center of mass.

00:13 Solid is a solid e, which is the tetrahedron, found at the plane x equals 0, y equals 0, z equals 0, and x plus y plus z equals 1.

00:30 And the density is row of x, y, z equals y.

00:37 First, let's find total mass.

00:43 Looking at our solid e, we see that x ranges from 0 to 1.

00:50 We integrate first from 0 to 1.

00:54 And then looking at the projection into the x, y, plane, we see that y range is from the line y equals 0 to the line y equals 1 minus x.

01:06 And now looking at a solid as a whole, you see that z ranges from z equals 0 to the plane z equals 1 minus x minus y and so this is the triple integral of our density function which is simply y d z d y d x and taking the antiderivative with respect to z and substituting we get integral from 0 to 1 integral from 0 to 1 minus x of and this is going to be y times 1 minus x minus y, dy, d .y, dx.

01:58 And taking the integer with respect to y, it says the integral from 0 to 1 of we have 1 half times 1 minus x times y squared minus 1 3rd y cube that i read it from 0 to 1 minus x d x in evaluating we get the integral from 0 to 1 to 1.

02:30 To 1 of 1 half times 1 minus x cubed minus 1 3 3rd times 1 minus x cubed d x which of course simplifies to 1 6th times the integral from 0 to 1 minus x cubed d x and taking the derivative with respect to x, we get one -sixth times negative one -fourth times one -minus to the fourth power from 0 to 1.

03:29 In evaluating, we get 1 -6th times 1 fourth, which is simply 1 -24th.

03:40 And this is our total mass.

03:46 To find the center of mass, first let's find the moment.

03:50 Of this solid.

03:52 We have the moment about the yz plane, miz.

03:59 This is our iterated integral from x equals 0 to 1, from y equals 0 to 1 minus x, and from z equals 0 to 1 minus x minus y, of our density function y times x...

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