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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 13-20, use a double integral to find the volume of the solid region.

Chapter 7.9, Problem 15E, Finding the Volume of a Solid Region In Exercises 13-20, use a double integral to find the volume of

To determine

To calculate: The volume of solid region that shown below,

Calculus: An Applied Approach (MindTap Course List), Chapter 7.9, Problem 15E

Explanation

Given information:

The provided solid region is

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 graph of solid region,

The equation of upper surface written as follow.

2x+3y+4z=12

Write the equation in function notation that is,

z=3x23y4

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

x-Coordinate y-Coordinate (x,y) Coordinate
0 4 (0,4)
6 0 (6,0)

The projected region R: 0x6,0y42x3 in the x-y plane plotted as follow,

The bounds for x are 0x6 and bounds for y are 0y42x3.

The volume for the solid region is,

Volume=06042x3(3x23y4)dydx

Now integrate with respect to y, and apply the limit

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