Explanation Given: The equation of the plane is x a + y b + z c = 1 . And the equations of coordinate planes are x = 0 , y = 0 , z = 0 Where, a > 0 , b > 0 and c > 0 . Formula used: The general point slope equation is y − y 1 = m ( x − x 1 ) . Calculation: Assuming the region R is formed when it is bound by x = 0 , y = 0 , z = 0 and x a + y b + z c = 1 . The region R is given as: Consider the x y − plane , Here, x lies between 0 to a . The slope of line AC is m . The value of m is calculated as: y 2 − y 1 = m ( x 2 − x 1 ) b − 0 = m ( 0 − a ) m = − b a Use the point slope method, which gives, y − y 1 = m ( x − x 1 ) y − b = − b a ( x − 0 ) y − b = − b a ( x ) a y + b x = a b Further, solve to get: y = b ( 1 − x a ) The limits of y are 0 to b ( 1 − x a ) and limits of x are 0 to a . Use the plane equation x a + y b + z c = 1 , which gives: z = c ( 1 − x a − y b ) Now, the limits of z are 0 to c ( 1 − x a − y b ) , limits of y are 0 to b ( 1 − x a ) and limits of x are 0 to a . Then, V = ∫ R d V = ∫ 0 a ∫ y = 0 b ( 1 − x a ) ∫ 0 c ( 1 − x a − x b ) d z d y d x = ∫ 0 a ∫ 0 b ( 1 − x a )

11th Edition

ISBN: 9780357092477

Author: Larson

Publisher: Cengage

See similar textbooks

Videos

Question

Chapter 14.2, Problem 66E

To determine

To calculate: What the volume of solid in the first octant bounded by the coordinates plane will be, where the plane xa+yb+zc=1.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.2, Problem 65E

Chapter 14.2, Problem 67E

Students have asked these similar questions

Calculus III May I please have the example, definition semicolons, and all blanks completed and solved? Thank you so much,

A company estimates that the revenue (in dollars) from the sale of x doghouses is given by R(x) = 12,000 In (0.02x+1). Use the differential to approximate the change in revenue from the sale of one more doghouse if 80 doghouses have already been sold. The revenue will increase by $ if one more doghouse is made. (Round to the nearest cent as needed.)

The population of bacteria (in millions) in a certain culture x hours after an experimental 20x nutrient is introduced into the culture is P(x) = - 2 Use the differential to approximate the changes in population for the following changes in x. 8+x a. 1 to 1.5 b. 3 to 3.25 a. Use the differential to approximate the change in population for x=1 to 1.5. Between 1 and 1.5 hours, the population of bacteria changes by million. (Round to three decimal places as needed.)

Chapter 14 Solutions

CALCULUS

Ch. 14.1 - CONCEPT CHECK Iterated Integral Explain what is...Ch. 14.1 - Prob. 2E Ch. 14.1 - Question: Evaluate the integral: 0x(2xy)dy Ch. 14.1 - Evaluating an IntegralIn Exercises 310, evaluate...Ch. 14.1 - Evaluating an IntegralIn Exercises 310, evaluate...Ch. 14.1 - Evaluating an IntegralIn Exercises 310, evaluate...Ch. 14.1 - Evaluating an Integral In Exercises 3-10, evaluate...Ch. 14.1 - Evaluating an IntegralIn Exercises 310, evaluate...Ch. 14.1 - Evaluating an Integral In Exercises 3-10, evaluate...Ch. 14.1 - Evaluating an Integral In Exercises 3-10, evaluate...

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.

What the volume of solid in the first octant bounded by the coordinates plane will be, where the plane x a + y b + z c = 1 .

Solution Summary: The author explains how to calculate the volume of solid in the first octant bounded by the coordinates plane.

Videos

To calculate: What the volume of solid in the first octant bounded by the coordinates plane will be, where the plane xa+yb+zc=1.

Want to see the full answer?

Chapter 14 Solutions