   Chapter 7.8, Problem 48E ### 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
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# Finding Area with Double Integral In Exercises 45–48, use a double integral to calculate the area denoted by ∫ R ∫ d A where R is the region bounded by the given equations. See Example 6. 2 x − y + 4 = 0 ,   2 x 2 − y = 0

To determine

To calculate: The area denoted by the expression RdA where R is the region bounded by the equations 2xy+4=0 and 2x2y=0 using double integral.

Explanation

Given Information:

The provided expression and the equations are RdA, 2xy+4=0 and 2x2y=0 respectively.

Formula used:

To determine the area in the plane by double integrals mostly two kind of strips are used.

1. Vertical strip: In case of vertical strip shown below the region bounded is given by,

axbg1(x)yg2(x)

And the area of the region is given by,

abg1(x)g2(x)dydx

2. Horizontal strip: In case of horizontal strip shown below the region bounded is given by,

cydh1(y)yh2(y)

And the area of the region is given by,

cdh1(y)h2(y)dxdy

Calculation:

Consider the expression RdA.

First sketch the graph of region R which is bounded by the equations 2xy+4=0 and 2x2y=0.

The graph of the equation 2x2y=0 would be a parabola.

For equation 2xy+4=0, put different values of x in the equation to obtain different values of y in order to plot graph of the equation.

For x=0,

y=2x+4=0+4=4

For x=1,

y=2x+4=2+4=6

For x=2,

y=2x+4=4+4=8

For x=3,

y=2x+4=6+4=10

Now construct a table showing different values of x and y as,

 x y=2x+4 0 4 1 6 2 8 3 10

Thus the graph of the equation 2xy+4=0 would be a straight line

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