00:01
Okay, so let's say you're driving in a car and your velocity profile is given by this equation here.
00:09
And you want to know the distance travel from time is equal to four seconds to time is equal to four seconds.
00:16
So how far do you get from four seconds starting from time of zero? well, to get that from a velocity, you just have to integrate the velocity, right? so x of t, x of 0, equal to the integral from 0 to the t, which should be a t here, a v of t d t.
00:41
Right.
00:43
So we have a v of t here.
00:45
So we just plug that in.
00:47
And the time we want is 4.
00:49
So we do x of 4 minus x of 0.
00:54
It's an integral from 0 to 4 seconds of our velocity, 3t, 16 minus t squared.
01:02
It's kind of a tricky integral, so we can use one of our integral tricks.
01:06
I'll be using a u sub.
01:08
So you set u is equal to 16 minus t squared.
01:13
Take the derivative of that, you get minus 2t, right? and then you plug in this u and this du into this equation here.
01:23
Here.
01:27
And that looks like this x of 4 minus x 0.
01:32
So you go to, we're going to keep these limits of integration for now, but just know that they're incorrect in terms of you at the moment.
01:41
Later on we're going to switch them back to we're going to solve the integral with respect to you and then switch the you used back to t's and then plug in 0 to 4.
01:51
So this looks like, so, the thing under the square root is just a u, right? but what about the 3t and the dt? well, to get that, we first have to isolate for...
02:03
Looks like t -d -t, right? oh, sorry, actually, this should be a d -t right here...