Hey, this is Presh Talwalkar
Solve for the area of the circle that is centered at the origin and
inscribed between the bell curves with equations y equals
e to the power of negative x squared and y equals negative e to the power of negative x squared
Can you figure it out? Give this problem a try and when you're ready keep watching the video for the solution
Let's first plot the curves
We have the positive bell curve y equals e to the power of negative x squared, and then we have the negative bell curve
Before I go over the correct approach let me explain an
invalid method
Because if I don't mention this someone will invariably leave a comment that they found an easier solution method
This approach will be a necessary condition, but not a sufficient condition
So this approach involves thinking about the geometry
If the circle is just touching the curves
Then you could then solve for the circle's radius
For the point at which it's tangent to the curves
This could be accomplished by setting the slope of the radius equal to the negative reciprocal of the slope or derivative of the curve
So what I'm going to explain is this is a necessary condition and
one of the values here will coincidentally get to the answer
but this is not a sufficient condition for
justifying the answer
Let me explain why?
One problem is that you might have the radius of the circle being completely vertical
This is because there's a point on this curve where the slope is zero
So if you were to consider the derivative of this curve you
May be overlooking the point at which the derivative is zero
Because the negative reciprocal would have to be an infinite value to consider the case where the radius is vertical
So one mistake is you might potentially be overlooking this solution
But even if you were to manually check for this case there would be another issue you
Would not know whether this circle is wholly contained between these bell curves
And it turns out this particular circle is actually not bound by these bell curves
So it would not be totally inscribed between these curves
Now if you were to use this incorrect approach you would actually find another solution which does correspond to the correct answer
but this will be an
Insufficiently justified answer because you might be overlooking other cases
And you would still have to show why the circle is completely inscribed between the curves
So this is not a valid approach to solving this problem
How can we solve it? Let's think about it geometrically. If
the radius of the circle is too small then it'll be wholly contained between the bell curves
But it won't be touching the curves at
Some point if we expand the radius the circle will be too large
So we're looking for the point at which the radius is just the right value
At this point the circles radius is just large enough to touch the bell curves
Any smaller radius would not touch the bell curves and any larger radius would not be inscribed between the curves
This point happens
When the radius is the minimal distance from the origin to the bell curves.
At this minimal value you can't have any smaller radius
And you also wouldn't want any larger radius because then you would be touching the bell curves at more points
And it would not be wholly inscribed
So let's use this approach
By symmetry we can solve for the distance to the positive bell curve with point x comma e to the power of negative x squared
So want to minimize the distance from the origin to the positive bell curve
And we can use the distance formula to get the following expression
But before we continue our calculations
Before we take the derivative and solve for when it's equal to zero there's actually an easier way to solve this problem
Since distance is a non negative value. We can more easily minimize the distance squared
This will be the distance squared from the origin to the positive bell curve
Which ends up being the same function without that messy square root, which will complicate our derivative and further calculations
This is a standard trick whenever we want to minimize the distance
So now let's tackle this problem. We want the minimum value of x squared plus e to the power of negative 2x squared
We'll take the derivative and set it equal to zero to solve for the critical values
We end up with the following equation which we can factor out 2x
And now we want to solve for when this equation is equal to zero so we have two possibilities
One is that 2x will be equal to zero which means X is equal to zero in?
That case we can check that the distance squared will be 1 is
This the minimal value
Well, let's check if the other solution leads to a smaller distance squared
So we have 1 minus 2 times e to the power of negative 2x squared
needs to be equal to 0
We can solve this equation to get x squared is equal to the natural log of square root of 2
If we substitute this back into the distance squared we end up getting the distance squared is approximately
0.847 and that's a smaller value than 1
So in fact this will be the minimal distance squared and this will be
the x value that we want. We want x squared equals the natural log of square root of two
we reject the solution of x equals zero, so
Now with this distance squared value we can then find that the
distance squared will be exactly the natural log of the square root of two plus one-half.
And notice the distance squared from the origin to the bell curves is also equal to the radius squared, r^2, of the circle
Because the distance is equal to the radius so finally we want the area of this circle
That'll be pi times r^2
which will equal pi times the distance squared so we have pi times the quantity
natural log of square root of 2 plus 1/2, and that's our answer
Did you figure it out?
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