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Chapter 13: Q. 35 (page 1055)

Describe the three-dimensional region expressed in each iterated integral:

∫−24 ∫26 ∫05 f(x,y,z)dydxdz

Short Answer

Expert verified

ℝ={(x,y,z)∣2≤x≤6,0≤y≤5,−2≤z≤4}

Step by step solution

01

Step 1. Given information.

We have been given a three-dimensional region:

∫−24 ∫26 ∫05 f(x,y,z)dydxdz

We have to describe it in the iterated integral.

02

Step 2. Evaluate.

By the definition of triple integral ∫a1a1 ∫b1b2 ∫c1c2 f(x,y,z)dzdydxrepresent the volume of the solid region ℝ=(x,y,z)∣a1≤x≤a2,b1≤y≤b2,c1≤z≤c2

The given triple integral represents the volume of the rectangular solid given byℝ={(x,y,z)∣2≤x≤6,0≤y≤5,−2≤z≤4}

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Most popular questions from this chapter

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

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

Assume that the density at each point in Ris proportional to the distance of the point from the xy-plane.

(a) Without using calculus, explain why the x- and y-coordinates of the center of mass are x¯=2andy¯=32,respectively.

(b) Use an appropriate integral expression to find the z-coordinate of the center of mass.

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

∫03∫01-y/3∫02-(2/3)y-2zf(x,y,z)dxdzdy

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.

∫∫Rx3ex2ydA,whereR={(x,y)|0≤x≤4and−1≤y≤1}

Use the results of Exercises 59 and 60 to find the centers of masses of the laminæ in Exercises 61–67.

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

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
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