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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 93-96, use a double integral to find the volume of the solid region.

Chapter 7, Problem 94RE, Finding the Volume of a Solid Region In Exercises 93-96, use a double integral to find the volume of

To determine

To calculate: The volume of the solid region shown below by application of double integral

Calculus: An Applied Approach (MindTap Course List), Chapter 7, Problem 94RE

Explanation

Given Information:

The provided solid region is shown below as;

Formula used:

To determine the volume of a solid follow the steps provided below as:

Step 1: First write the equation of the surface in form,

z=f(x,y)

Which is followed by sketching of solid region

Step 2: Next sketch the region R in the two dimensional x-y plane and then find the order and limits of integration.

Step 3: Finally solve the obtained double integral,

Rf(x,y)dA

By using the necessary information from step 2.

The integration of function f(x)=xn,n1 is;

xndx=xn+1n+1+c

Calculation:

Consider the provided solid region.

The equation of the surface is 2x+4y+3z=24

First write the equation in the z=f(x,y) form as;

2x+4y+3z=243z=242x4yz=823x43y

To sketch the graph in the x-y plane eliminate the z plane, so the equation for x-y plane would be,

2x+4y=244y=242xy=242x4y=612x

Next sketch the graph of region R in the x-y plane shown below as,

By observation of the above graph it is clear that the region R is bounded by the equations y=0,x=0 and y=612x.

To set the double integral select x as the outer variable.

Thus, x is outer variable of integration with constant bounds 0x12 and the bounds for y would be 0y6(12)x

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