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Finding the area between two circles inner-tangential to a parabola.

Imagine a valley y=x^2.
A circle with radius = 5 is inside it.
Another circle, with radius = 4 is above the previous circle, (and inclined to right side)

(PLEASE SEE THE ATTACHMENT zip file)

Find the area between y=x^2 and the 2 circles

(Not allowed to use computer.)

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We call the centre of the circle with radius 5, O and the one with radius 4, O'. Also let's call the point where y=x^2 is tangent to the circle r=5, B and the point where it is tangent to r=4, C. Please write the names of the points on the shape before going to the next step.
Now, we know that OB=5 is perpendicular to the curve y=x^2 at B, because y=x^2 is tangent to the circle at that point and the radius at that point will make the right angle. For the same reason O'C is prependicular at C. Therefore, we can distinguish a trapezoid with two right angles. The ...

Solution Summary

The area between two circles inner-tangential to a parabola is

$2.19