![]() One and a minus one half so we can simplify them a little bit. And then this term right over here squared is going to be positive one fourth X to the negative four. Product of these two and then multiplied that by two, yep, it's just gonna be negative one half. And so it's going to be negative one half. X to the negative two which is going to be one. These times two is going to get us negative, Let's see X squared time Term right here squared is going to be X to the fourth over four. Out what is one plus F prime of X squared? So it's going to be, So one, one plus F prime of X, F prime of X squared is going to be equal to it's gonna be one plus this thing squared. Now, what is F prime of X squared? So it's going to be, Actually let's just write And so that's going toīe negative one half X to the negative two. This is one half X to the negative one is one way to think about it. Over six is going to be X squared over two, and The derivative of that is three X squared. ![]() So what is F prime of X going to be? Let's see, X to the third, Need to square it, we need to add one to it, and then we X we just need to figure out what F prime of X is, we Of X we could actually deal in terms of other variables. Lower boundary in X to the upper boundary in X,Īnd this is the arc length, if we're dealing in terms We got arc length, arc length is equal to the integral from the We got a justification for the formula of arc length. The arc length formula correctly, it'll just beĪ bit of power algebra that you'll have to do toĪctually find the arc length. So I encourage you to pause this video and try it out on your own. And so we've already highlighted that in this purple-ish color. Length along this curve between X equals one and X equals two. And what I want to do in this video is to figure out the arc Imagine the center of a square, then trying to take it's "arc length" as you would a circle, it doesn't work, because a square is a bit different in it's construction and it's properties are different.- This right here is the graph of Y is equal to X to the third over six plus one over two X. ![]() It does not extend because these arcs don't have circular properties. not a straight line (though equally aplicable if you wished). ![]() This arc length problem involves non circular arcs, it is talking about "curve length", i.e. This conversion would give the accurrate answers: 180 degrees * 30 pi / 360 = 15 pi. This also means we changed the definition for our radians, since 1 degree is now actually equal in length to: Degree = 1/360 * 30pi radians). In our case we changed this value to C = 30 pi. The unit circle is based on a circle with radius of 1, therefore it's diameter it's full circumference is 2pi (it has a diameter of 2, i.e. But our circumference should be 15pi, not just pi. Using our above formula won't simply work: 180 * 2pi / 360 = pi. This logically means an arc with angle measure 180 degrees would have a length of 15pi units (it is half of the circle). Imagine we had a much larger circle, with diameter 30. This is only for the unit circle however. 45 degrees, and multiplying it by 2pi / 360 to get the radian measure which would be pi/4 radians in this case. We also know that a circle has 2pi radians in it (convertible as 2pi = 360 degrees, thus 1 degree = 2pi / 360 radians) This is where you're getting that, you taking the degree, i.e. This is because the circumference of a circle is C = pi * d Yes, the arc length you're talking about is changing a circles degree measurement into a measure.
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