Graph the curve with parametric equations $x=\cos t$
$y=\sin 3 t, z=\sin t .$ Find the total length of this curve
correct to four decimal places.
all right, we would have find the length of the curve that's represented by these three Parametric equations. So one of the things that we can do is write this in vector form as our of tea, which we can represent as Chris a nifty sign of three t and sign of tea. Using our vector notation, we want to take the derivative of the spectre with respect to t. So for our first term, we end up with negative sign of tea For our second, we end up with three co sign of three t and for the third, we end up with co sign of tea. Then I want to find the magnitude of that derivative vector. So we're gonna take the magnitude which is going to be, uh, remember, this is the square root of the sum of our squared component functions. So, by square, negative sign of T I'm going to end up with sine squared of tea by square My 2nd 1 I'm gonna end up with nine co sine squared of three t I square my third. I'm gonna end up with co sign squared of tea. Well, that's gonna be into the square root time and we know thinking about our trick identity is that sine squared plus co sine squared is always equal toe one So we can write this as one plus nine co sine squared three t and then to find the length of that function or of that, uh, curve, we have to integrate the function that we just found in terms of tea. We're gonna integrate with respect to T. But we need to think about what are interval is going to be. And if we look at the functions that we have all of our component functions, they're all signs and co signs. Onda. We know that whenever we plug input into, um into our science and co signs time and co sign functions, the interval 0 to 2 pi is going to give us unique values. But then after that, um, we're going to start seeing the same values again so we can consider this on the interval from zero to two pi. So we're gonna go ahead and evaluate brat using our graphing calculator. It's clear everything out. We're gonna go to our why equals menu, and we're gonna put our function in there so we have square root of one plus nine times the CO sign of three T or three X In the case of our calculator, all squared name and then integrate. We're gonna go to our math menu, Scroll down option nine, which is are integral, definite, integral. We're gonna go from zero to to pie. And you want to make sure that your calculator is in radiance mode and not degree modes will check that go to mode we see in our third row that we are in terms of radiance and not degrees. That's good. So you've got 0 to 2. Pi is our balance of immigration, and we're gonna put our function. Why one in there and integrate with respect to t or again X is the calculator represents it and we're gonna click Enter This one takes just a moment or two and now we have the evaluation of this integral. So working around this to four decimal places and what we'll get is 13.9744