 Welcome back everyone. What I want to do in this example is actually show how we can use the disk method to prove standard Volume formula from geometry. We're actually going to compute the volume of a cone You might remember from your previous high school or college geometry class I mean, I guess maybe if you're really on top of things you saw this in middle school Or who knows maybe middle school or elementary school one day, right? Who knows whatever? Anyways How do we get all these volume formulas like a step volume of a sphere is four-thirds? What is it four-thirds pi r cubed? I hope I said that one right the volume of a cone We're gonna see in a moment is one third pi r squared times h and you know You can talk about pyramids and literally this volume form has come from it turns out We're gonna actually talk about some of these In our discussion of volume for right now. I just want to talk about the volume of a cone So if we think of a cone Think about like the type of ice cream cone you've eaten before right look at the volume of something that looks like this Right and so the parameters of the cone that we're going to be considering are the two following The radius of the cone is Significant because the wider the base the more volume have the shorter the base the less volume have also the height of the cone This right here is going to be of significance. We're going to call the height h in the radius r So we can actually realize that the a cone is a solid of revolution If we take this right triangle we now see highlighted in red and you spin that around The axis of the cone that that the rotation of that triangle forms the cone So cones are just solids of revolutions formed by rotating Right triangles and so you can see the following to the left right here Let's let's treat this as a solid revolution type problem If we think of a typical cross-section of this triangle you see here this this rectangle right here Let's orient it so that the axis of the cone actually coincides with the x-axis and then the cross-sectional Rectangles are going to be perpendicular to the x-axis and we're going to spin those around the x-axis forming our disc right here Then the height of our rectangle here is if this point at the top of the rectangles x comma y The height of the rectangle I should say the the radius of the cylinder It's going to be this y-coordinate and So we're looking at the volume of our cone will be the integral of pi y squared DX DX is the thickness of the of the disc Why is the radius of the disc now where? How is the x-coordinate allowed to change well if you look at the extremes right here? One extreme is you get zero zero right here x could equal zero so x equals zero right there On the other extreme where can we do because the the rectangle can live anywhere in this region right here? oops and So the rectangles can go anywhere in this region The other side would be over here when x is actually the height of the Cone and the y-coordinate still be zero so x is going to range from zero to h Now we have to deal with this y-coordinate right because this y-value That would be great if we're integrating with respect to y But the thickness of the rectangles DX so we have to integrate with respect to x how do you write this this? Y-coordinate in terms of the x-coordinates right well the idea is to use the slope of the line to help you out here Because if we focus on this line right here It's just a line. We can find the line by slope intercept form So we get this y equals mx plus b We've positioned it so that the apex of the cone goes to the origin So that's going to tell us that the y-coordinate is just a zero So our line is going to look like y equals mx We need to find the slope of this line and we can find slope of a line using the standard slope formula M equals rise over 1 y 2 minus y 1 over x 2 minus x 1 But we have to know two points on the line one point We can use the origin the other point we want to use is right here and this is going to be a point which is h is It's x-coordinates y-coordinate. It's going to be r. So basically we're going to run a similar triangle type argument here So you're going to end up with r minus zero Over h minus zero so your slope is r over h and so you didn't see captured In our diagram the label right here y equals r over h x And so we're going to plug that in For the y-coordinate y equals r over h x So if we put that in there I'm going to factor the pi out of the integral because it's just a constant zero to h We're going to end up with an r squared over h squared times x squared dx I'm also going to take the r squared h squared out because again This is just a constant so we can take it out and so we end up with The expression pi r squared over h squared Integral from zero to h of x squared dx And so now as we integrate this by the power rule I'm just going to copy the pi r squared over h squared again When you take the using the power rule here x squared is going to become an x cubed over three Go from zero to h You're going to plug in the h and get an h cubed when you plug in zero you'll just get zero and so we end up with a pi r squared h cubed um all over Let's see three h squared right there And so now the thing to consider is we'll simplify the h's there's h squared on bottom That cancels with two of the h's on the top and that'll then leave us with a pi r squared h and then there's just there's just a three that's left in the numerator Excuse me the denominator there and so if i were to just tweak this a little bit We get one third pi r squared h cubed the usual The usual error of volume formula for acone And so this can be derived using this disc method or for more general You might want to use the washer method and so this this technique of calculating volumes can be extremely useful To not just establishing the standard volume formulas of solids. We've seen in previous geometry courses, but also for many irregular solids as well And so in the next video, we're actually gonna in the next lecture. I should say which is another video, of course We're going to be looking at some generalizations of this washer and disc method. We've seen here in lecture four So take a look for those videos to learn some more about these how we can use integration to find volumes of three-dimensional solids here If you do like the videos you're watching feel free to like the video post a comment If you have any questions, for example, also comment below. I'd be happy to answer those For you if you didn't understand anything or just want some more clarification on the videos right here Subscribe to see some more videos like this in the future and I'll see you next time. Bye everyone