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Home Multiple Integrals Infinite Sum Theorem and Volume Examples Example 3
 
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Example 3

Find the volume of the solid bounded by the plane

z = 2y

and the paraboloid

z = 1 - 2x2 - y2.

12_multiple_integrals-175.gif

Figure 12.3.8 (a): Surfaces

12_multiple_integrals-176.gif

Figure 12.3.8. (b): Region D

Step 1

The surfaces and the region D are sketched in Figure 12.3.8.

Step 2

The two surfaces intersect on the curve

2y = 1 - 2x2 - y2,

or solving for y, y = - 1 + 12_multiple_integrals-177.gif Therefore D is the region

-1 ≤ x ≤ 1, -1-12_multiple_integrals-178.gif ≤ y ≤ -1 + 12_multiple_integrals-179.gif

Step 3

We see from the figure that the plane is the lower surface and the paraboloid is the upper surface.

12_multiple_integrals-180.gif

12_multiple_integrals-181.gif

= 12_multiple_integrals-182.gif=12_multiple_integrals-183.gif

V =12_multiple_integrals-184.gif

Put x = sin θ, 12_multiple_integrals-185.gif= cos θ, dx = cos θ dθ (Figure 12.3.9).

12_multiple_integrals-187.gif

12_multiple_integrals-186.gif

Figure 12.3.9

Answer:

V = √2π.
Home Multiple Integrals Infinite Sum Theorem and Volume Examples Example 3

Last Update: 2006-11-15