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Chapter 13: Q 28 (page 1039)

Let Tbe triangular region with vertices (0,0),(1,1),and(1,-1)

If the density at each point in Tis proportional to the point鈥檚 distance from the x-axis, find the mass of T.

Short Answer

Expert verified

Mass of triangular region ism=k3.

Step by step solution

01

Given Information

We need to determine mass of triangular region.

Density at each point is proportional to point's distance fromxaxis

蚁(x,y)=ky

02

Calculation of Mass

Mass is symmetric about xaxis.

Hence, total mass is twice of mass of upper half triangle.

鈬�m=2鈭�01鈭�0xkydydxm=鈭�惟蚁(x,y)dA

Solving inner integral first

m=2k鈭�01y220xdx

m=k鈭�01x2dx

m=kx3301

Solving further

m=k3

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Most popular questions from this chapter

Let f(x, y, z) and g(x, y, z) be integrable functions on the rectangular solid R=x,y,z|a1猢�x猢�a2,b1猢�y猢�b2,c1猢�z猢�c2. . Use the definition of the triple integral to prove that :

鈭�Rf(x,y,z)+g(x,y,z)dV=鈭�Rf(x,y,z)dV+鈭�Rg(x,y,z)dV.

Use Definition 13.4to evaluate the double integrals in Exercises 29鈥�32.

鈭�Rxy3dA

whereR={(x,y)鈭�-2鈮�x鈮�2&-1鈮�y鈮�1}

Evaluate the sums in Exercises 23鈥�28.

鈭�j=13鈭�k=12kj

In the following lamina, all angles are right angles and the density is constant:

Evaluate Each of the integrals in exercises 33-36 as an iterated integral and then compare your answer with thoise you found in exercise 29-32

鈭�鈭�Rxy3dAWhereR={(x,y)|-2鈮�x鈮�2,-1鈮�y鈮�1}
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