Explanation Given: Given planes are x = 0 y = 0 z = 0 x a + y b + z c = 1 Concept used: Volume of tetrahedron = ∫ ∫ ∫ d z d y d x To evaluate a triple integral, first z variable is integrated and then y variable is integrated and at last x variable is integrated. Calculation: x a + y b + z c = 1..... ( i ) ⇒ z c = 1 − ( x a + y b ) ⇒ z c = 1 − x a − y b ⇒ z = c ( 1 − x a − y b ) Limits of z : 0 to c ( 1 − x a − y b ) To, get limits of y put z = 0 in equation (i), x a + y b = 1 ⇒ y b = 1 − x a ⇒ y = b ( 1 − x a ) Limits of y : 0 to b ( 1 − x a ) To get limits of x put y and z equal to zero in equation (i), x a = 1 ⇒ x = a Limits of x : 0 to a Volume of tetrahedron is: ⇒ ∫ ∫ ∫ d z d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ∫ 0 c ( 1 − x a − y b ) d z d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ( ∫ 0 c ( 1 − x a − y b ) d z ) d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ( ∫ 0 c ( 1 − x a − y b ) d z ) d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ( [ z ] 0 c ( 1 − x a − y b ) ) d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ( c ( 1 − x a − y b ) − 0 ) d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ( c ( 1 − x a − y b ) ) d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ( c − c x a − c y b ) d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) c − c x a − c y b d y d x ⇒ ∫ 0 a ( ∫ 0 b ( 1 − x a ) c × 1 − c x a × 1 − c b × y d y ) d x ⇒ ∫ 0 a ( [ c y − c x a × y − c b × y 2 2 ] 0 b ( 1 − x a ) ) d x ⇒ ∫ 0 a ( c [ { b ( 1 − x a ) } − x a × { b ( 1 − x a ) } − 1 b × { b ( 1 − x a ) } 2 2 ] ) d x ⇒ c ∫ 0 a (

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

See similar textbooks

Videos

Question

Chapter 12, Problem 26PE

To determine

To find the volume of tetrahedron that is bounded by the coordinate planes and the plane xa+yb+zc=1

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12, Problem 25PE

Chapter 12, Problem 27PE

Students have asked these similar questions

The graph of f' is below. Use it to determine where the local minima and maxima for f are. If there are multiple answers, separate with commas. 2 f'(x) N -5 -4 3-2-1 -1 -2 -3 -4 12 3 4 5 -x Local minima at x Local maxima at x

The graph of f' is below. Use it to determine the intervals where f is increasing. -5-4-32 4- 3 2 1 -2 -3 +x 2 3 4 5

The graph of f' is below. Use it to determine where the inflection points are and the intervals where f is concave up and concave down. If there are multiple inflection points, separate with a comma. 6 5 4 3 2 1 f'(x) +x -6-5-4-3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6+ Inflection point(s) at x = Concave up: Concave down:

Chapter 12 Solutions

Calculus

Ch. 12.1 - Prob. 1PS Ch. 12.1 - Prob. 2PS Ch. 12.1 - Prob. 3PS Ch. 12.1 - Prob. 4PS Ch. 12.1 - Prob. 5PS Ch. 12.1 - Prob. 6PS Ch. 12.1 - Prob. 7PS Ch. 12.1 - Prob. 8PS Ch. 12.1 - Prob. 9PS Ch. 12.1 - Prob. 10PS

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To find the volume of tetrahedron that is bounded by the coordinate planes and the plane x a + y b + z c = 1

Solution Summary: The author explains how to find the volume of tetrahedron that is bounded by the coordinate planes and the plane.

Videos

To find the volume of tetrahedron that is bounded by the coordinate planes and the plane xa+yb+zc=1

Want to see the full answer?

Chapter 12 Solutions