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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 7.9, Problem 3E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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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 = 8 − 8 x − 2 y

To determine

To calculate: The volume of solid region bounded in first octane and the plane z=8−8x−2y.

Explanation

Given information:

The provided equation of plane is z=8−8x−2y.

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=∫mn∫abf(x,y)dxdy

Here, the projected region is R:

m≤y≤n and a≤x≤b

Calculation:

Consider equation of plane,

z=8−8x−2y

The following table shown different coordinate of (x,y,z) for z=8−8x−2y.

x-Coordinate y-Coordinate z-Coordinate (x,y,z) Coordinate
0 0 8 (0,0,8)
0 1 0 (0,1,0)
4 0 0 (4,0,0)

The graph of function z=8−8x−2y in first octant shown below.

Substitute, 0 for z in equation of plane z=8−8x−2y.

8−8x−2y=08x+2y=8y=4−4x

The following table shown different coordinate of (x,y) for y=4−4x.

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

The graph of equation y=4−4x is

One way to set up the double integration is to choose x as the outer variable with that choice. The bounds for x are 0≤x≤1 and bounds for y are 0≤x≤4−4x and another choose to set up the double integration is to choose y as the outer variable

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Sect-7.1 P-1CPSect-7.1 P-2CPSect-7.1 P-3CPSect-7.1 P-4CPSect-7.1 P-5CPSect-7.1 P-6CPSect-7.1 P-7CPSect-7.1 P-1SWUSect-7.1 P-2SWUSect-7.1 P-3SWU
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