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Chapter 5: Q19P (page 247)

Above the square with vertices at, (0,0), (2,0),(0,2) and (2,2) and under the plane z = 8-x+y.

Short Answer

Expert verified

The required solution is 32.

Step by step solution

01

Definition of double integral

The double integral of f(x,y)over the areaA in the (x,y)plane as the limit of this sum, and we write it as∫∫Af(x,y)dxdy.

02

Calculation of the volume under the curve

The volume above the square with vertices at (0,0),(2,0),(0,2) and (2,2), and under the plane z=8-x+y.

First, integrate over z:

|=∫x=02∫y=02∫z=08-x+ydzdxdy=∫02dy∫02dx8-x+y

03

Further calculation of the volume of the bounded region

Now, integrate over x:

|=∫02dy8x-12x2+yx02=∫02dy(16-2+2y)

04

Further calculation of the volume of the bounded region

Now, integrate over y:

|=∫02dy(16-2+2y)=(14y+y2)02=32

Therefore, the value is 32

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Most popular questions from this chapter

Prove the “parallel axis theorem”: The moment of inertia of a body about a given axis is l=lm+Md2, where M is the mass of the body,lmis the moment of inertia of the body about an axis through the center of mass and parallel to the given axis, and dis the distance between the two axes.

A lamina covering the quarter disk x2+y2≤4,x>0,y>0 has (area) density . Find the mass of the lamina.

Under the surface z = 1 /(y+2) , and over the area bounded by and y=x y2+x=2.

For a square lamina of uniform density, findabout

(a) a side,

(b) a diagonal,

(c) an axis through a corner and perpendicular to the plane of the lamina. Hint: See the perpendicular axis theorem, Example 1f.

Prove the following two theorems of Pappus: The areainside a closed curve in the (x , y) plane, y≥0 , is revolved about the x axis. The volume of the solid generated is equal to times the circumference of the circle traced by the centroid of A. Hint: Write the integrals for the volume and for the centroid.
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