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In this section we discussed examples of ordinary, everyday functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function.
verse indefinite, integral us and the difference between each This is gonna be the very basic beginnings of integral. So the first thing we're gonna do before we really get into definite birth indefinite is some of the properties of integral roles that are going to be useful to know. So here, some properties of in a gross. If you have the integral from a tow A of a function, it's going to be zero. If you have the integral from A to B and then you have some constant C times of function, you can pull that constant out of the integration like so, if you have the integral of A to be of a function ffx plus a function G of X, we can move this integral through the plus or minus. So you would have a to b of ffx waas a to B. Jeeva. So this one, this last one is pretty important, um, that you can move an integral signed through pluses and minuses and you can also pull the constants out. So again, these air the property. So now we'll go ahead and talk about what I mean by definite and indefinite. So starting with a definite integral, a definite integral. It's going to take the form of A to be, which are we're going to refer to as your bounds. And then we're gonna have a function and then d of X so ffx. This is the function that needs to be integrated. X represents your variable of integration. Everything else A, B and F FX are based off of this variable of integration. It's going to be very important. It could be dx dy y. Do you? Those are all very common ones that we're going to use. So your variable of integration determines what the rest of your integral looks like and a and B those are are bound. So I'm going to call those bounds A and B need to depend on this variable of integration. X. So when you do this definite integral, you're actually going to do the integration and then plug in A and B into your answer. So when you integrate a definite integral, you're gonna end up with a number an actual number. You should have no variables in your answer. When you integrate a definite integral, we should have no variables. However, when we talk about indefinite, integral roles. So for an indefinite in a role, we're going to have the form. It's just gonna be the integral of f of X DX. F of X is still the function that needs to be integrated. X is still your variable of integration. All the property still remain the same. Integration techniques remain the same. However, your answer to an indefinite integral will be a function. So you will have some function that still has variables in it, plus a constancy. You must add this constancy when you have an indefinite integral. This is very, very important. So definite, integral. You'll get a number indefinite. Integral will be a function. And you need to make sure you add this constancy to the end now. So we're going to start with the integral from 0 to 3 of X squared DX. So to perform integration, we're gonna add one to the power which gives us X Q and we're going to multiply by the reciprocal of whatever is in that power. So the reciprocal of three is one third. We're going to bring that out front. We're going to evaluate this at three, and that zero so we have one third times three cute minus one third times zero Cute. This is zero. It's going to go away three times. Siri's 99 times Siri's 27 27/3 gives us the final answer of nine. So the integral from 0 to 3 of X squared is equal to nine. Another example, The integral from 1 to 2 of six x dx constant six can be pulled out from 1 to 2 of x dx. We have six times. This is a power of one. So we add one to the power and give get X squared. Reciprocal of two is one half evaluated from two and one six over to is three so we can do three X square. So three times to squared minus three times one squared two times two is 44 times three is 12 minus three. Again, we should get a final answer of nine. The integral of five x cubed E x. So here the first thing to notice is that we don't have bound, so this is going to be an indefinite integral, but the integration techniques stays the same. So we're gonna pull the five out we have the integral of X cubed DX. We add one which gives us X fourth in the power multiply by the reciprocal which is 14 So this means we have 5/4 X to the fourth. We have no bounce to plug in. We just simply add the plus c constant. This is the final answer of the indefinite. And okay, Another example. We have the integral of X plus two X squared DX. We can see another property of integration here. We can move this integral through the plus sign and break this into two intervals. You can just integrate right now or you could break it up. It's a personal preference. So the integral of X integrates to one half X squared, plus add one. You can pull this constant out if you like. I'm going to show it to you guys not being pulled out. So we have the two. We add one which gives us execute, and then we have to multiply by one third. So we have one half x squared, plus two thirds execute and then we have to add R plus C constant because we have an indefinite integral giving us this the final answer for this type of integral