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Chapter 13: Q. 25 (page 991)

If the density at each point in T is proportional to the point's distance from the y-axis, find the mass ofT.

Short Answer

Expert verified

The mass of the triangular lamina is

m=23k

Step by step solution

01

Given information

The objective of this problem is to find the mass of the triangular region.

02

calculation

For an example, take a triangle with vertices (0,0),(1,1) and (1,-1).

Mass of the lamina is can be calculated by the formula

m=∬Ωρ(x,y)dA

ρ(x,y)is the uniform density of the lamina. Ifρ(x,y)is proportional to the point's distance from the y - axis.

ρ(x,y)=kxm=∫01∫-xxkxdydx

Integrate with respect to y first.

m=∫01kx[y]-xxdxm=∫01kx[x-(-x)]dxm=2k∫01x2dx

Integrate with respect to x.

m=2kx3301

Substitute the limits

m=23k

Thus, the mass of the triangular lamina is

m=23k

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Most popular questions from this chapter

Identify the quantities determined by the integral expressions in Exercises 19–24. If x, y, and z are all measured in centimeters and ρ(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.

∭Ω zρ(x,y,z)dV

Evaluate the triple integrals over the specified rectangular solid region.

∭R (x+2y+3z)dV, whereR={(x,y,z)∣0≤x≤4,1≤y≤5,and2≤z≤7}

In Exercises 61–64, let R be the rectangular solid defined by R=(x,y,z)|0≤a1≤x≤a2,0≤b1≤y≤b2,0≤c2≤z≤c2.

Assume that the density of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

Let Ωbe a lamina in the xy-plane. Suppose Ωis composed of n non-overlapping laminæ role="math" localid="1650321722341" Ω1,Ω2,…,Ωn.Show that if the masses of these laminæ are m1,m2,…,mnand the centers of masses are x¯1,y¯1,x¯2,y¯2,…,x¯n,y¯n,then the center of mass of Ωis x¯,y¯,where

x¯=∑k=1nmkx¯k∑k=1nmkandy¯=∑k=1nmky¯k∑k=1nmk.

Evaluate each of the integrals in exercise 33-36 as iterated integrals and then compare your answers with those you found in exercise 29-32

∫∫Rx2y3dAWhereR={(x,y)|1≤x≤3,0≤y≤2}
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