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Chapter 14: Q. 6 (page 1131)

Give three examples of quantities that may be computed by鈭�R鈥�G(x,y)dA.

Short Answer

Expert verified

Ans: The three examples of quantities that may be computed by 鈭�R鈥�G(x,y)dA.are:

1) Surface area is computed by the double integral.

2) The mass of lamina with density function 蚁(x,y,z)is computed by double integral.

3) The volume under the surface z=zx,yis computed by the double integral.

Step by step solution

01

Step 1. Given information.

The objective is to provide three examples of quantities that may be computed by 鈭�R鈥�G(x,y)dA.

02

Step 2. First example

The surface area is computed by the following double integral:

鈭�R鈥�鈭�z鈭�x2+鈭�z鈭�y2+1dA.

In this case,

G(x,y)=鈥�鈭�z鈭�x2+鈭�z鈭�y2+1.

03

Step 3.  The second example

The mass of lamina with density function 蚁x,y,zis computed by following double integral:

鈭�R鈥�蚁(x,y,z(x,y))鈭�z鈭�x2+鈭�z鈭�y2+1dA

In this case,

localid="1651053939013" G(x,y)=鈥�蚁(x,y,z(x,y))鈭�z鈭�x2+鈭�z鈭�y2+1.

04

Step 4. The third example

The volume under the surfacez=zx,y is computed by following double integral:

鈭�R鈥�z(x,y)dA

In this case

G(x,y)=鈥�z(x,y)

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Most popular questions from this chapter

Find the masses of the lamina:

The lamina occupies the region of the hyperbolic saddle with equation z=x2-y2that lies above and/or below the disk of radius 2 about the origin in the XY-plane where the density is uniform.

Integrate the given function over the accompanying surface in Exercises 27鈥�34.
f(x,y,z)=xyz2, where S is the portion of the cone 3z=x2+y2 that lies within the sphere of radius 4 and centered at the origin.

F=xln(xz),5z,1y2+1 , where S is the region of the plane with equation 12x鈭�9y+3z=10, where 2鈮�x鈮�3and 5鈮�y鈮�10, with n pointing upwards.

Let Rbe a simply connected region in the xy-plane. Show that the portion of the paraboloid with equation z=x2+y2 determined by R has the same area as the portion of the saddle with equation z=x2-y2determined by R.

Problem Zero: Read the section and make your own summary of the material.
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