   Chapter 7.9, Problem 2E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
1 views

# Finding the Volume of a Solid Region In Exercises 1-6, find the volume of the solid region bounded in the first octant by the plane. See Example 1. z = 6 − x − 3 y

To determine

To calculate: The volume of solid region bounded in first octane and the plane z=6x3y.

Explanation

Given information:

The provided equation of plane is z=6x3y.

Formula used:

The procedure to calculate volume of surface z=f(x,y),

Step-1: Write the equation of surface in the form z=f(x,y)

Step-2: Sketch the projected region R in the x-y-plane.

Step-3: Determine the order of limits of integration.

Step-4: Evaluate the volume of solid region,

Volume=mnabf(x,y)dxdy

Here, the projected region is R:

myn and axb

Calculation:

Consider equation of plane,

z=6x3y

The following table shown different coordinate of (x,y,z) for z=6x3y.

 x-Coordinate y-Coordinate z-Coordinate (x,y,z) Coordinate 0 0 6 (0,0,6) 0 2 0 (0,2,0) 6 0 0 (6,0,0)

The graph of function z=6x3y in first octant shown below.

Substitute, 0 for z in equation of plane z=6x3y.

6x3y=0x+3y=6

The following table shown different coordinate of (x,y) for x+3y=6.

 x-Coordinate y-Coordinate (x,y) Coordinate 6 0 (6,0) 0 2 (0,2)

The region R is bounded by the line x=0,y=0 and x+3y=6.

One way to set up the double integration is to choose x as the outer variable with that choice. The bounds for x are 0x6 and bounds for y are 0y6x3 and another choose to set up the double integration is to choose y as the outer variable. with that choice, The bounds for y are 0y2 and bounds for x are 0x63y.

The volume for the solid region is,

Volume=0606x3(6x3y)dydx

Now integrate with respect to y, and apply the limit

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Find more solutions based on key concepts 