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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 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.

y = x + 6 , y = x 2

To determine

To calculate: The area denoted by the expression RdA where R is the region bounded by the equations y=x+6 and y=x2 using double integral.

Explanation

Given Information:

The provided expression and the equations are RdA, y=x+6 and y=x2 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 y=x+6 and y=x2.

The graph of the equation y=x2 would be a parabola.

For equation y=x+6, 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=x+6=0+6=6

For x=1

y=x+6=1+6=7

For x=2

y=x+6=2+6=8

For x=3

y=x+6=3+6=9

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

x y=x+6
0 6
1 7
2 8
3 9

Thus the graph of the equation y=x+6 would be a straight line

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