   Chapter 7.P, Problem 15P

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# The circle with radius 1 shown in the figure touches the curve y = | 2 x | twice. Find the area of the region that lies between the two curves.

To determine

To Find:

The area of the region that lies between the two curves.

Explanation

Calculations:

Slope of the Perpendicular line segment is 12

Equation of the line with slope 12 is given by y=12x+b;b1

Equation of the circle with radius 1 is given by x2+y2=1x2+y2=1x2+(12x+b)2=15x24bx+4(b21)=0

Since the tangent line will touch the circle at one place only, the equation of the circle will have only one root.

Hence, the discriminant of the quadratic equation is zero

We have 5x24bx+4(b21)=0Δ=0(4b)24.5.4(b21)=0b2=54b=52

The upper limit of integration is obtained by solving the quadratic equation for b=52

We have 5x225x+1=0(5x1)2=0x=15

Required Area is given by

A=2015(5x21x2)dx=015(5x</

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