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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

The figure shows a region consisting of all points inside a square that are closer to the center than to the sides of the square. Find the area of the region.

FIGURE FOR PROBLEM 18

Chapter 5, Problem 18P, The figure shows a region consisting of all points inside a square that are closer to the center

To determine

To find: The area of the region.

Explanation

Given information:

A region consisting of all points inside a square that are closer to the center than the sides of the square.

Calculation:

Consider R be the region consisting of all points inside a square that are closer to the center than to the sides of the square.

Sketch the region consisting of all points inside a square that are closer to the center than the sides of the square as shown in Figure 1.

Refer to Figure 1.

Side length is L=2.

The line y at the top side of the square is y=1.

The distance from the origin to the points inside a square that are closer to the center is 1y.

The points inside the square are located equal distance from the center of the circle and the top side of the square.

The region 18R is noted in Figure 1.

Provide the equation of the circle as shown below.

x2+y2=(1y)2x2=1+y22yy2x2=12y (1)

Get equation of the parabola from Equation (1).

y=1x22

The region bounded by the parabola y=1x22 on the left by the y-axis x=0 and below by the line y=x.

Find the intersection point of the parabola where the line y=x as shown below.

Substitute x for y in Equation (1).

x2=12xx2+2x=1x2+2x1=0

Find the value of x as shown below.

x=b±b24ac2a=2±44×1(1)2=2+222=21

Find the area of the region as shown below.

A=8021(yx)dx

Substitute 1x22 for y in the above Equation

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