 Hello and welcome to a screencast about finding the average value of a function. So we have a kind of definition here for us, but it says if f of is a continuous function on the interval from a to b, then it's average value. So that's what we're gonna be looking at today. On that interval from a to b is given by the formula 1 over b minus a times the integral from a to b of f of x dx. So this integral here is what we've been looking at recently, and that's the area under the curve. We first used Riemann sums to approximate this area, but then we also can use some geometry to figure out what this area is exactly. Okay, then this also consists of an interval. So this b minus a here is actually the length of our interval, okay? All right, so just to kind of get an idea of where these pieces come from. So that way you can see that. All right, so here's a function g of x. This should look very familiar from a previous screencast. So it's kind of a funky function, but we have some lines and then we also have a semi-circle here. So they're all pieces that we can actually figure out what the area is. So remember again, this average value function is defined to be the area under the curve. So we need to figure out what that value is first. So that's gonna be all of this area here, this negative area down here, and then this positive area over here under the circle. So in a previous screencast, you calculated that area. I'll just roll through really quickly what that answer was again. Let me get rid of all my scribbles so you can see things. Yeah, so if you already know what these answers are, that's fantastic. Go ahead and pause the screencast or fast forward if you want to, to figure out then what this area is. So we broke this integral up into three pieces, or no, I'm sorry, more than three pieces. One, two, three, four pieces. We're gonna go from zero to two of our function g of x dx. Then we're gonna go from two to three of our function. And then we're gonna do the integral from three to six of our function. And then we're gonna do the integral from six to eight of our function. And the good news is then this gives us the overall integral from zero to eight of our function, which is really what we want. Okay, so that's our one piece of our average value function that we need. Okay, so as you roll through some of these values, let's see. So we have from three to two, so we have a rectangle here and then a triangle here on top. So this gives us a value of five plus, let's see, here we've got a really skinny triangle that gives us a value of five halves. And then we have another triangle down here. Let's see the width of this base is three and the height is negative two. So that gives us an area of negative three. And then finally we have our semicircle, so that has a radius of one. If you were to split that diameter in half, so that would have an area of pi over two. Cuz it's half of the circle and the radius is one. Okay, so anyway, these are all of our different values here. So when we total these guys up, and I just realized this one is not correct, this is not five. I thought that seemed kind of small. Let's see, so we do three times two, which is six plus two times two. So that would be, that's eight, okay? I apologize for that. I had a five in the brain cuz I was doing probably the eight and the negative three in my head already. That's what I get for working ahead, right? Okay, so anyhow, when you go ahead and figure these values out, let's see if we've got some fractions in here that we can work with. So that's gonna give us five, which will be 10 over two. Plus five over two plus pi over two. So that's gonna give us a grand total of 15 plus pi all over two. Okay, so again, this was the area that was given to you in the previous screencast. But now we need to add to it the average value, okay? So then we're gonna need to multiply by one over the length of our integral. So this is our interval, this is our area. Sorry, I didn't get myself enough room here. So my average value is gonna be one over eight minus zero times my integral from zero to eight g of x dx. Which is gonna give me one eighth times this 15 plus pi over two. So putting these two fractions together, we multiply. So we're gonna get 15 plus pi all over 16. And that would be the average value then of this function. Okay, all right, the other graph you guys looked at then on a previous screencast was this function here h of x. So again, we wanna figure out the average value of this function on the interval from zero to six. So we're gonna break this again up into three integrals. So if we look at the integral from zero to six of, oops, our function is h this time, not g, sorry about that, h of x dx. That's gonna be the integral from zero to two, cuz that's a fairly nice shape of h of x dx. Plus the integral from two to four of h of x dx, cuz again, that's a nice shape. And then finally, the integral from four to six of our function h of x dx. Cuz again, that's a nice shape. Okay, so these nice shapes, so now we need to figure out the areas of them. So from zero to two, we have a rectangle, and that has a base of two and a height of three, so that gives us an area of six. From two to four, we have a triangle, and that has a base of two and a height of three, so that's gonna have an area of three. And then from four to six, we have a negative area because it's below the x axis, so that has a base of two and a height of negative three, so that's gonna have an area of negative three. So that's gonna give us a grand total of our area to be six, okay? But now we want the average value. So the average value of this function on the interval is one over six minus zero times the integral from zero to six of our function h of x dx. So that gives us one six times six, which gives us a grand total of one. All right, thank you for watching.