 In this video, we're gonna find the length of the arc of the semi-cubicle parabola y squared equals x cubed between the points one, one and four, eight. And now, yes, you can double check in your dictionary here. Semi-cubicle is, in fact, a real word, we didn't just make it up. And it's describing this curve right here, y squared equals x squared. Now, if we wanna go from the point one, one to four, eight, as we see right here, notice this only requires the upper half of our semi-cubicle parabola. So what we actually can do for simplicity in this state is we're gonna solve for y and get the square root of x cubed or more simply, this is gonna be x to the 3 1⁄2 power. Now, be aware that normally when you take a square root, there should be two options, plus or minus. The plus does choose the top half, the negative chooses the bottom half. We don't need the negative half here. We just need the top half because the one, one and the four, eight show up on the top half here. So this is gonna be a function that we're gonna use in this calculation here. So notice that if we go from one, one, that's where x equals one to four, eight, that's when x equals four. If we wanna find the length of this arc, it makes sense to use the arc length formula we had learned before. So for example, s equals the integral of ds, ds, for which case we've seen previously that this is equal to the integral of the square root of dx squared plus dy squared. Now, factoring out the dx squared, we can put this in a form that's gonna be more useful for us. We can take the integral of the square root of one plus f prime of x squared dx. And so this is the form that we're gonna need to calculate this thing right here. So our integral, some of these things we already know, we're integrating with respect to x. We need to know the bounds of x, which we already determined that x will range from one to four. We're gonna get the square root of one plus, well, we need the derivative of our function. So notice here, y, that was x to three halves. So by the power rule y prime, we equal three halves x to the one half. And now we need to square that thing. Squaring the three halves will give us a nine fourths and then squaring the square root actually gives us an x. Like so, in which case we get a dx right here. One thing you're gonna notice from this example is that we chose the semi-cubicle parabola exactly for the reason that when you put it into the arc length formula, this turns into a very doable calculation. We can find the anti-derivative here. And we're gonna do a u substitution. Since we have a linear function inside of the square root, that'll be our u, the linear function, one plus nine fourths x. Taking the derivative, that'll be a nine fourths dx. So we need a nine fourths right here inside the integral. So we'll times by four ninths to correct that. And also changing the bounds, right? So as we switch from x to u, we have to worry about the numbers four and one. Plugging those in there, if you plug four in there, nine fourths times four is just nine plus one is a 10. That's not so bad. If you plug in a one, you're gonna get one plus nine fourths, which that's the same thing as four fourths plus nine fourths. So you're gonna get 13 fourths as we change the bounds. So then our integral becomes four ninths, the integral from 13 fourths to 10. And then we're gonna get the square root of u, or that is u to the one half power du. Oops, that u is running away from us. So we get this, which this is not so bad as it's a function to find its antiderivative. By the power rule, we raise the power back up to this city cubicle parabola, right? So you increase the power by one u, we'll go up to three halves power. We need to divide by three halves, which is the same thing as times by two thirds. So you'll times by the reciprocal. And we need to plug in 13 fourths and a 10. Bring this up a little bit more. So multiplying the coefficients together, we're gonna end up with eight over 27. And then plugging in the 10, we're gonna get 10 to the three halves minus 13 to 13 fourths to the three halves, like so. Now be aware that taking something to three halves power u to the three halves, this is the same thing as just taking u times the square root of u. And so I think that's how we're gonna finish this thing up eight over 27, we're gonna get 10 root 10. And then we're gonna get minus 13 root 13 over four root four. Now notice four is a perfect square. The square root of four is two. So four times two is of course an eight. Got a little head of myself there. If we distribute the eight through to kind of clean up the fractions a little bit, you're gonna have a 127th out in front. And then you're gonna end up with eight times 10, which is 80, 80 root 10 minus, well eight will cancel the eight in the denominator. So you get 13 screwed 13, like so. And again, you can distribute the one over 27 through if you want to, but this gives us the exact answer that we were looking for. And this is sort of an interesting looking irrational number. If we wanna know exactly what it adds up to be, you know, estimate this using a calculator of some kind, scientific calculator can handle this just fine. And this will be approximately 7.633705. And so that was kind of an interesting calculation that we accomplished right there. Now, if we go back and look at the graph of this thing, which was above, there it is. What if we were looking at just the straight line that connects these two points, right? You can see that the straight line itself is pretty close. I mean, there is curvature to it, right? But it's pretty close here. And if you use the distance formula just between the two points, you're gonna get the square root of 58 as the distance between them. And this is gonna be approximately 7.615773, which if you compare that two, can I get them on the same view? Probably not, 7.6. This one was also 7.6. This one's a little bit bigger than the square root of 58. And that's because again, we are measuring arc, there's a curve to it. Now, this calculation wasn't so bad, but honestly, the semi-cubical parabola was created for basically one purpose in mind. It is the best function for arc length calculations. And so this is its moment to shine. Unfortunately, arc length calculations generally don't turn out to be as easy as the semi-cubical parabola turned out to be here. And we'll see some other examples in just a moment.