Chapter 13: Q 70. (page 1016)

Let $a, c$ be positive real numbers. Prove that the volume of the pyramid with vertices $(- a, a, 0), (a, - a, 0), (- a, - a, 0), and (0, 0, c) is \frac{4}{3} a^{2} c$ .

Short Answer

Expert verified

Use type I integral to prove the above result.

Step by step solution

Given Information

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

$a, c$ are real positive numbers.

Using Concept of Type I Integral

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

For type I integral, $- a \leq x \leq a, - a \leq y \leq a$

Since the volume of pyramid with vertices $(0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c)$ is $\frac{1}{6} a b c$

If pyramid has square base as $b \equiv 4 a and a \equiv 2 a$

Hence, volume is

$\frac{1}{6} (2 a) (4 a) c = \frac{4}{3} a^{2} c$

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Let $a, c$ be positive real numbers. Prove that the volume of the pyramid with vertices $(- a, a, 0), (a, - a, 0), (- a, - a, 0), and (0, 0, c) is \frac{4}{3} a^{2} c$ .

Short Answer

Step by step solution

Given Information

Using Concept of Type I Integral

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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

Step by step solution

Given Information

Using Concept of Type I Integral

One App. One Place for Learning.

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Let $a, c$ be positive real numbers. Prove that the volume of the pyramid with vertices $(- a, a, 0), (a, - a, 0), (- a, - a, 0), and (0, 0, c) is \frac{4}{3} a^{2} c$ .