00:01
An application of antiderivative is finding the area of the region bounded by curve.
00:10
So in this problem, we're given with or we're asked to find the area of the region bounded by the curve y equals x squared over 4 minus 4 and the x -axis.
00:23
So what we will do first is to sketch the graph so as to see the region and the boundaries.
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Now, y equals x squared over 4 minus 4 is a parabola that is shifted down 1 by 4.
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So we find a vertex right here.
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To find the intersection point with the x -axis, we have to take y equal to 0.
00:48
So if y is equal to 0, then we have 0 equals x squared over 4 minus 4.
00:58
We add both sides of the equation by 4, we get 4 equals x squared over 4.
01:03
And we multiply both sides by 4, giving us x squared equals 16, which if we take the square root, turns into plus and minus 4.
01:11
Therefore, the intersection points are at x equals negative 4 and x equals 4.
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So we're going to plot this.
01:22
Then we have the curve, something, we have a curve that looks like this.
01:26
It should be a parabola opening upward and extending in both positive and negative direction.
01:33
So our region here is this one shaded with blue.
01:39
So the area for this one, we get by considering a very small strip first.
01:46
Let's say this.
01:46
So for the area of the rectangular strip, it's obtained by the very small area we call da, is the length which is defined by the equation y up minus y down.
02:03
That's the length.
02:07
And then multiplied by the small thickness, which is dx.
02:13
So the equation at the top is the x -axis, which is y equals 0...