Chapter 13: Q 69. (page 1016)

Let $a, b, c$ be positive real numbers. Prove that the volume of the pyramid with vertices $(0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c)$ is $\frac{1}{6} a b c .$

Short Answer

Expert verified

It can be proved by solving $\iint_{Ω} f (x, y) d y d x$ .

Step by step solution

Given Information

A pyramid has vertices $(0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c)$

$a, b, c$ are real positive numbers.

We need to prove that volume is $\frac{1}{6} a b c$

Solving for type I integral

The equation of line lying in cartesian plane is

$y - 0 = \frac{b - 0}{0 - a} (x - a)$

$y = - \frac{b}{a} x + b$

Region of integration is bounded by lines $y = - \frac{b}{a} x + b, x = 0, y = 0$

For type I integral $0 \leq x \leq a, 0 \leq y \leq - \frac{b}{a} x + b$

Evaluating Volume

Volume is given by

$\iint_{Ω} f (x, y) d y d x = \int_{0}^{a} \int_{0}^{- \frac{b}{a} x + b} f (x, y) d y d x$

Function of pyramid by equation of plane is given by

$f (x, y) = z = c - \frac{c}{a} x - \frac{c}{b} y$

It gives

$\int_{0}^{a} \int_{0}^{- \frac{b}{a}} f + b (x, y) d y d x = \int_{0}^{a} \int_{0}^{- \frac{b}{a}} (c - \frac{c}{a} x - \frac{c}{b} y) d y d x$

Simplification

Evaluation of double integral gives

$\int_{0}^{a} \int_{0}^{- \frac{b}{a}} (c - \frac{c}{a} x - \frac{c}{b} y) d y d x = \int_{0}^{a} {[c y - \frac{c}{a} x y - \frac{c}{2 b} y^{2}]}_{0}^{\frac{- b}{a} x + b} d x$

$= \int_{0}^{a} [c (- \frac{b}{a} x + b) - \frac{c}{a} x (- \frac{b}{a} x + b) - \frac{c}{2 b} {(- \frac{b}{a} x + b)}^{2}] d x$

$= \int_{0}^{a} (- \frac{b c}{a} x + \frac{b c}{2 a^{2}} x^{2} + \frac{1}{2} b c) d x$

It implies

$\int_{0}^{a} \int_{0}^{- \frac{b}{a}} (c + \frac{c}{a} x - \frac{c}{b} y) d y d x = {[- \frac{b c}{2 a} x^{2} + \frac{b c}{6 a^{2}} x^{3} + \frac{1}{2} b c x]}_{0}^{a}$

$= - \frac{1}{2} a b c + \frac{1}{6} a b c + \frac{1}{2} a b c$

$= \frac{1}{6} a b c$

Hence volume is $\frac{1}{6} a b c$

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Let $a, b, c$ be positive real numbers. Prove that the volume of the pyramid with vertices $(0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c)$ is $\frac{1}{6} a b c .$

Short Answer

Step by step solution

Given Information

Solving for type I integral

Evaluating Volume

Simplification

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Let a,b,cbe positive real numbers. Prove that the volume of the pyramid with vertices(0,0,0),(a,0,0),(0,b,0),(0,0,c)is16abc.

Short Answer

Step by step solution

Given Information

Solving for type I integral

Evaluating Volume

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Discrete Mathematics

Decision Maths

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Let $a, b, c$ be positive real numbers. Prove that the volume of the pyramid with vertices $(0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c)$ is $\frac{1}{6} a b c .$