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Welcome back. We’re doing the last part of problem one of the 2008 calculus BC exam and I’ll repeat the problem. It says the region R models the surface of a small pond at all points in R at a distant x from the y axis. The depth of the water is given by h of x is equal to three minus x. Find the volume of the water in the pond. So we thought about it, this is you know a prospective drawing of this. This is the surface of the pond right here and I was trying to do draw the depth, so when x is equal to zero the pond is exactly three I guess you know we can say feet or whatever the units are deep and then when x is equal to two the pond is one unit deep and so to find the volume we think about it the same way. What is the cross section at any point of the pond look like? Well let's see, if we were to take a cross section, so let's take this cross section of the pond and let me do it in different color and that could be kind of confusing. That’s not too different, so that’s green.
So the height along the surface of the pond is going to be the difference between the two functions and we know that the top function is sin of pi x and we know that this bottom function is x to the third minus four x. So let me see if can draw this cross section so we notice the width along the surface of this cross section is going to be the difference between this function and this function, right. So this is that same cross section. The width along the top I know I'm confusing you, is going to be this function minus this function so it's going to be, sin of pi x minus x to the third minus four x and what's going to be the height right here. Well, they told us the height of the pond or the depth of the pond at any point is three minus x whatever x value are at.
So this is going to be three minus x. So the area of this cross section of the pond is going to be sin of pi x minus x to the third minus four x times the depth of the pond. That’s the surface width times the depth. So that times three minus x, so that’s the area of each cross section. So if we want the volume for the entire pond we take essentially the volume of each of this kind of slivers of the pond. So we take the area of each cross section and we multiply it by a very small width to get a very small sliver. So you take that and multiply it by dx and we integrate. We sum up all of the slivers of the pond from x is equal to zero to x is equal to two. So let's do that, let's integrate from x is equal to zero to x is equal to two.
Once again this is really hard integral. It's something that you can do especially if you use integration by parts but it's a little bit messy and you only have 45 minutes to do all three problems on the BC exam so I'm assuming that they want you to use the calculator here. So let's use the calculator to evaluate it. So let’s see, so that’s, this was part C that we already have in our calculator. We can actually do second enter and we’ll get the previous entry, second enter. So we already typed in a bunch of stuff. The only difference between what we did in part C and now instead of having in part C we had this expression squared. Now we don’t have a squared but we’re multiplying it by times three minus x so let's do that.
So if we go here we can delete this squared. We’re not squaring it anymore, delete it and now let’s do so let's do second insert so that’s insert, times three minus x close parenthesis and so let's see we have sign of pi x minus x to third plus four x.
That’s the same thing as this times three minus x and are variable integration is x and were integrating from x is equal to zero to x is equal to two, let's hit enter. And you notice that it is calculating and our answer is 8.37. I think it's a fair answer. So the volume this time, the volume of the pond is equal to 8.37. Anyway, hopefully you found that useful. I will everyday try to do a couple of these problems. See you in the next video.