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Chapter 13: Q 70. (page 1016)

Let a,cbe positive real numbers. Prove that the volume of the pyramid with vertices(-a,a,0),(a,-a,0),(-a,-a,0),and(0,0,c)is43a2c.

Short Answer

Expert verified

Use type I integral to prove the above result.

Step by step solution

01

Given Information

The vertices of pyramid are (a,a,0),(-a,a,0),(-a,-a,0),(0,0,c)

a,care real positive numbers.

02

Using Concept of Type I Integral

The region of integration 惟is bounded by y=-a,y=a,x=a,x=-a

For type I integral, -a鈮�x鈮�a,-a鈮�y鈮�a

Since the volume of pyramid with vertices (0,0,0),(a,0,0),(0,b,0),(0,0,c)is 16abc

If pyramid has square base as b鈮�4aanda鈮�2a

Hence, volume is

16(2a)(4a)c=43a2c

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Most popular questions from this chapter

Let f(x)be an integrable function on the rectangular solid R=x,y,z|a1猢�x猢�a2,b1猢�y猢�b2,c1猢�z猢�c2, and let 魏鈭�鈩�.Use the definition of the triple integral to prove that:

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