 Hello everyone and welcome to this lesson on finding volumes of solids by using cross sections. There's a really great website. I'll be taking you to that will give you a great visualization of what it is. We're talking about you will see the URL there on the PowerPoint slide. You might also hear this topic talked about as finding volumes of solids by slicing. The analogy I like to use think of slicing a loaf of bread and you're trying to find the volume of that loaf of bread. So if you were to slice it up and find the volume of each individual slice and then add those volumes together, you would have the volume of the entire loaf of bread. That's basically what we're talking about here. So let me take you to the website. If you want to explore this later on your own, it's the bottom section parallel cross-section figure. So let's start with this one. Let me stop it right there for you. Now you'll notice at the bottom you have a base shape. That shape will be given to you created by equations that will be noted in the problem. What you're trying to do then is take cross-sections of certain shapes. In this case, it's an equilateral triangle. So if you notice the base of the equilateral triangle sits on that base shape of the solid and then it rises up from there. Here you're talking about three-dimensional shapes now. You have an x-axis, a y-axis and now a z-axis. That is the part coming up. Think of it out of the paper if you want to think of it that way. You'll notice that because of the shape of the cross-section that influences the basic shape of the solid overall. So in this case, it almost looks pointy at the top. So let me keep running this for you. So there's your base shape that looks like an ellipse and notice how this it's kind of pointy at the top almost like a helmet of sort and notice also that because of that shape of the ellipse and therefore the length of the base of that equilateral triangle cross-section. Notice how the size of the cross-section changes as you move across that base shape. So right now the equilateral triangle is kind of small. But notice it rises as it gets to towards the middle then decreases in size again. So let's go look at another shape. Let's try this one where our cross-sections are squares. Notice it's the same base shape for your solid. This time know the cross-sections are squares. Let me stop it for you right there. Again, notice that the size of that cross-section square changes as you move across the base shape. Again, because the length of that side of the square is changing. But notice how because now the cross-section is a square, notice how the shape of the solid overall has changed from how when it was an equilateral triangle. So let's try one more. The other one I like to do is the circles one because that's one you could see. Notice the same base shape. This time know your cross-sections are circles and this generates a solid that looks like an egg or a football. However you want to think of it. Notice how the cross-sections essentially cut through the the the center of the the base shape. All right, notice how it's the diameter of the circle that lies along the length of that base shape. So once again, the size of your circles changes as you move across that base shape. And that's really what creates that egg shape to the solid overall. So the question then becomes how do you do this? Well think of what you've learned so far. Remember that when you've learned about area under a curve we first talked about Riemann sums and the idea of using rectangles under a curve finding the area of each individual one and adding them up. And remember that turned into definite integrals as a means to find the definite area under the curve. You're going to do something similar here. So let's talk about then how we're going to use definite integrals to actually help us calculate these volumes. Once again, you're going to have the issue of your representative rectangles either being vertical or horizontal. If they are vertical and your cross sections you are told are perpendicular to the x-axis, that's going to make it a DX problem. And therefore, of course, your A and B values, your limits of integration are going to be x values as well. You are integrating the area of the type of cross section. So let's talk for a second about the types of cross sections you're mostly going to see. On this slide, you see the most common ones. Squares, obviously the length of your representative rectangle becomes the base of that cross sectional square. So to get the area of that, you simply do the side, the base squared. Rectangles, you will be told the height of the rectangle. The representative rectangle creates the base length of the rectangle, and then you'll be told the height by which to multiply. Semi-circles, one-half pi r squared, of course, is the formula. The big thing you need to remember is the radius calculation. The length of your representative rectangle is the diameter of the semi-circle. So you need to cut that in half first before squaring it. And isosceles triangle is simply going to be your usual area of a triangle formula. One-half the base, and remember the base length is the length of your representative rectangle, multiplied by the height. And finally, equilateral triangles is one that you'll see. You might remember that we saw one in the demonstration with the applet. The area of an equilateral triangle, of course, is square root of 3 over 4 times the base length squared. Think of it as it's almost a proportional part, a fractional part of the area of a square. It's square root of 3 fourths of the area of the square. So there's some of the common ones you're going to see as you go about doing the problems. So when we talk about taking the integral of the area of the type of cross-section you have, we're talking about using those area formulas you just saw. For cross-sections taken perpendicular to the y-axis, it then becomes a dy problem. Your limits of integration then are y-values. Your area that you're going to be integrating you need to remember must be in terms of y. So that might be some rearranging of an equation you'll have to do. So the basic process is a three-step simple one. The first thing you're going to want to do is come up with an expression for the length of the base of the type of cross-section you're told you have. You're going to use that representative rectangle idea and to come up with your expression for the length of that rectangle. You will use that top minus bottom or right minus left idea we've talked about before. Secondly, you're going to find an expression for the area of the type of cross-section that you're working with and those are these on this previous slide. Finally, you're going to set up your integral for finding the volume. You can either evaluate it by hand by finding the anti-derivative and evaluating that way or you are free to use your calculator to do that evaluation.