   Chapter 12.1, Problem 46E

Chapter
Section
Textbook Problem

# Find the volume of the solid that lies inside both of the spheres x2 + y2 + z2 + 4x − 2y + 4z + 5 = 0 and x2 + y2 + z2 = 4

To determine

To find: The volume of the solid that lies inside both of the spheres x2+y2+z2+4x2y+4z+5=0 and x2+y2+z2=4 .

Explanation

Consider a sphere with center C(h,k,l) and radius r.

Formula:

Write the expression for an equation of a sphere with center C(h,k,l) and radius r.

(xh)2+(yk)2+(zl)2=r2 (1)

Here,

(h,k,l) is the center of a sphere and

r is the radius of a sphere.

Rewrite the equation of sphere x2+y2+z2+4x2y+4z+5=0 as follows.

x2+4x+y22y+z2+4z+5=0(x2+4x+22)+(y22y+12)+(z2+4z+22)+(5221222)=0(x+2)2+(y1)2+(z+2)2+(5414)=0(x+2)2+(y1)2+(z+2)2=4

[x(2)]2+(y1)2+[z(2)]2=22 (2)

By comparing equation (2) with equation (1), the center and radius of the sphere x2+y2+z2+4x2y+4z+5=0 is (2,1,2) and 2 respectively.

Rewrite the equation of sphere x2+y2+z2=4 as follows.

x2+y2+z2=4

(x0)2+(y0)2+(z0)2=22 (3)

By comparing equation (3) with equation (1), the center and radius of the sphere x2+y2+z2=4 is (0,0,0) and 2 respectively.

As the radius of both spheres is same, the volume inside both spheres is symmetrical about the plane that contains the circle of intersection of the spheres.

The two spheres are drawn as shown in Figure 1.

Write the expression to find the distance between two points P1(x1,y1,z1) and P2(x2,y2,z2)

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