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Home Multiple Integrals Infinite Sum Theorem and Volume Examples Example 2
 
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Example 2

Find the volume of the solid bounded by the four planes x = 0, y = 0, z = x + y, z = 1 - x - y.

Step 1

Sketch the planes. We see from Figure 12.3.7 that z = x + y is below

z = 1 - x - y

Step 2

Find inequalities for the region D. Since the two planes

z = x + y, z = 1 - x - y

meet at the line

2x + 2y =1, y = ½ - x,

D is the region

0 ≤ x ≤ ½, 0 ≤ y ≤ ½ - x.

Step 3

V = 12_multiple_integrals-169.gif

= 12_multiple_integrals-170.gif

= 12_multiple_integrals-171.gif

12_multiple_integrals-172.gif=12_multiple_integrals-173.gif

= ½ - x - 2x(½ - x) - (½ - x)2 = ¼ - x + x2.

12_multiple_integrals-174.gif

12_multiple_integrals-168.gif

Figure 12.3.7
Home Multiple Integrals Infinite Sum Theorem and Volume Examples Example 2

Last Update: 2006-11-15