the-volume-of-the-region-between-the-planes-x-y-2z-2-and-2x-y-z-4-in-the-first-octant-is-

Question Number 210723 by universe last updated on 17/Aug/24

the volume of the region between the

planes x + y + 2 z = 2 and 2 x + y + z = 4 in

the first octant is

Answered by mr W last updated on 17/Aug/24

V = \frac{1}{6} (2 \times 4 \times 4 - 2 \times 2 \times 1) = \frac{14}{3}

Commented by universe last updated on 17/Aug/24

s i r c a n u e x p l a i n y o u r m e t h o d l i t t l e b i t

a n d h o w t w s o l v e b y i n t e g r a t i o n m e t h o d

Commented by mr W last updated on 17/Aug/24

Commented by mr W last updated on 17/Aug/24

t h e e q n . o f b l u e p l a n e i s

\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1

t h e v o l u m e u n d e r t h e p l a n e i s

V_{1} = \frac{a b}{2} \times \frac{c}{3} = \frac{a b c}{6}

s i m i l a r l y t h e v o l u m e u n d e r t h e

r e d p l a n e \frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 1 i s

V_{2} = \frac{p q r}{6}

t h e v o l u m e b e t w e e n b l u e p l a n e a n d

r e d p l a n e i s

V = V_{1} - V_{2} = \frac{1}{6} (a b c - p q r) .

i n t h i s e x a m p l e t h e r e d p l a n e i s

x + y + 2 z = 2, i . e . \frac{x}{2} + \frac{y}{2} + z = 1

a n d t h e b l u e p l a n e i s

2 x + y + z = 4, i . e . \frac{x}{2} + \frac{y}{4} + \frac{z}{4} = 1 .

t h e r e f o r e

V = \frac{1}{6} (2 \times 4 \times 4 - 2 \times 2 \times 1) = \frac{14}{3}

Commented by mr W last updated on 18/Aug/24

Commented by mr W last updated on 18/Aug/24

t h e r e i s n o n e e d t o a p p l y i n t e g r a t i o n .

Commented by universe last updated on 18/Aug/24

t h a n k u s i r