Welcome back. We’re doing the last part of Problem 1 of the 2008 Calculus BC exam.
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AP Calculus BC exams – 2008 1 d AP Calculus BC
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 distance x from the y-axis, the depth of the water is given by h of x is equal to 3 minus x. Find the volume of the water in the pond.
So we thought about, this is, you know, a perspective drawing of this, this is the surface of the pond, right here. And I was trying to draw the depth. So when x is equal to 0, the pond is exactly 3, I guess, you know, we could say feet, or whatever the units are, deep, and then when x is equal to 2, the pond is 1 unit deep.
And so to find the volume, we think about it the same way. What does a cross section at any point of the pond look like? Well if we, let’s see. So if we were to take a cross section, so let’s say that this, let’s take this cross section of the pond.
Let me do it in a different color, I know it can 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 sine of pi x, and we know that this bottom function is x to the third minus 4x. So if I were to draw, let me see if I can draw this cross section. So we know this the height, sorry, the width along the surface of this cross section is going to be the difference between this function and this function, right? So if I were to draw, let me see if I can, so this is that same cross section, it’s the width along the top, I know I’m confusing you, is going to be this function minus this function.
It’s going to be sine of pi x minus x to the third minus 4x. 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 3 minus x, whatever x value we’re at.
So this is going to be 3 minus x. So the area of this cross section of the pond is going to be sine of pi x minus x to the third minus four x times the depth of the pond, right? That’s the surface width times the depth. So that times 3 minus x.
So that’s the area of each cross section. So we want the volume for the entire pond, we take essentially the volume of each of these 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 we take that and multiply it by dx, and we integrate. We sum up all of these slivers of the pond from x is equal to 0 to x is equal to 2. So let’s do that.
Let’s integrate from x is equal to 0 from x is equal to 2. Once again, this is a really hard integral. Or 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 this was part c that we already have in our calculator. We could do second enter, and we’ll get the previous entry.
Second enter. Because we already typed in a bunch of stuff. The only difference between what we did in Part C and now is, now, instead of having in Part C, we had this expression squared. Now, we don’t have it squared, but we’re multiplying it by times 3 minus x.
So if we go here, we can delete this squared. We’re not squaring it anymore. Deleted. And now let’s do, I want you to see my key strokes.
So let’s do second insert. So let’s insert times 3 minus x close parentheses. And so, let’s see. We have sine of pi x minus x to the third plus 4x, right, that’s the same thing as this.
Times 3 minus x. And our variable of integration is x, and we’re integrating from x is equal to zero to x is equal to 2. Let’s hit enter.
And it noticed, it is calculating. And our answer is 8.37, I think, is 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, every day, try to do a couple of these problems. See you in the next video.
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