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Chapter 13: Q. 40 (page 1055)

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

鈭02鈭01-3x/2鈭2x+(4/3)y-40f(x,y,z)dzdydx

Short Answer

Expert verified

The three-dimensional region is given by planer equation,

x2+y3-z4=1

Step by step solution

01

Step 1. Given Information 

We are given,

鈭02鈭01-3x/2鈭2x+(4/3)y-40f(x,y,z)dzdydx

02

Step 2. The three dimensional region. 

By the definition of triple integral 鈭a1a1鈭b4b2鈭a1c2f(x,y,z)dzdydxrepresent the volume of the solid region鈩=(x,y,z)鈭a1鈮x鈮a2,b1鈮y鈮b2,c1鈮z鈮c2

Using this definition, we get

The given integral 鈭02鈭01-3x2鈭2x+43y-40f(x,y,z)dzdydxrepresents the volume of the tetrahedron whose vertices are(0,0,0)(2,0,0)(0,3,0)(0,0,-4)

Since from the given integral

z=2x+43y-4鈬6x+4y-3z=12x2+y3-z4=1

From this planer equation the vertices of the tetrahedron becomes (0,0,0)(2,0,0)(0,3,0)(0,0,-4).

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Most popular questions from this chapter

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 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炉=2andy炉=32,respectively.

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

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

鈭鈭Rxx+ydA,whereR={(x,y)|1鈮x鈮4and0鈮y鈮3}

In Exercises, let S=(x,y)鈭x2+y2鈮1andx鈮0.

If the density at each point in S is proportional to the point鈥檚 distance from the origin, find the center of mass of S.

Explain how to construct a Riemann sum for a function of two variables over a rectangular region.

Evaluate the triple integrals over the specified rectangular solid region.

鈭R鈥ln鈦xyz2dV, whereR={(x,y,z)鈭1鈮x鈮3,1鈮y鈮e,and1鈮z鈮2}
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