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Chapter 13: Q 39. (page 1039)

Let Rbe rectangular region with vertices (0,0),(b,0),(0,h),and(b,h)

If the density at each point in Ris proportional to the point鈥檚 distance from the y-axis, find the mass of R.

Short Answer

Expert verified

The mass ofRism=12khb2.

Step by step solution

01

Given Information

The vertices of rectangular region are (0,0),(b,0),(0,h)and(b,h)

蚁(x,y)=kx

02

Calculation of Mass

The formula is m=鈭�惟蚁(x,y)dA

m=鈭�0b鈭�0hkxdydxas 蚁(x,y)=kx

Solving inner integral first m=鈭�0bkx[y]0hdx

m=鈭�0bkxhdx

m=kh鈭�0bxdx

m=khx220b

m=12khb2

The required mass ism=12khb2

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Most popular questions from this chapter

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

鈭�鈭�Rsin(x+2y)dAWhereR={(x,y)|0鈮�x鈮�蟺,0鈮�y鈮�蟺2}

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

鈭�03鈭�01-y/3鈭�02-(2/3)y-2zf(x,y,z)dxdzdy

In Exercises 57鈥�60, let R be the rectangular solid defined by

R = {(x, y, z) | 0 鈮� x 鈮� 4, 0 鈮� y 鈮� 3, 0 鈮� z 鈮� 2}.

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.

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

鈭�惟鈥�z蚁(x,y,z)dV

Expressthesum25+26+27+28+45+46+47+48usingdouble-summationnotation.
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