The region enclosed by the given curves is rotated about the specified line. Find the volume of

The region enclosed by the given curves is rotated about the...

Key Concepts

Volume of Revolution

This concept involves finding the volume of a three-dimensional solid formed when a two-dimensional region is rotated around an axis. It combines integration and geometry, allowing the use of methods like the washer or shell methods to sum up infinitesimal volume elements.

Washer Method

The washer method is used to compute the volume of a solid of revolution when the solid has a hole in the center. It involves integrating the difference between the outer radius squared and the inner radius squared multiplied by ?, over the interval defined by the region's boundaries.

Setting Up the Integral

This process requires identifying the limits of integration by determining where the given curves intersect, and expressing the radii (either outer or inner) in terms of the variable of integration. In the case of rotation about the y-axis, the x-coordinate expressions become the radii in the integral setup.

Transcript

00:01 I want to find the volume and i'm looking at the equation x equals 2y minus y squared, as well as x equals zero.

00:09 So the area that's bounded here is this part i'm coloring in because x equals zero we know is that y -axis.

00:15 But i know it's really tough to graph an x equals equation, especially when maybe i don't have a reference of what it looks like.

00:21 So i did want to show over here one way that even just now as i sketched it out, i went through and drew the graph.

00:28 And i just picked numbers to plug in for y and then just be really aware as you're plotting them, you either want to, you know, rewrite it as x comma y as your coordinate or just remember that you're plotting the y value first as your height and then finding the x.

00:44 So i just picked easy things to plug in for y and then just kind of quickly calculated that.

00:49 You know, if i fill in zero for both, then my answer for x is still zero.

00:52 If i fill in one for both y's, and i just went through and found those points.

00:56 So it can kind of help me draw that graph.

00:59 So talking again about volume, i'm looking about that part i shaded in in glue, and we're going to rotate this around the y -axis.

01:07 So the y -axis is that vertical line, x -equal zero.

01:12 Well, the y -axis, there is no area that i need to take away.

01:18 There's no volume i need to take away as this rotation is happening.

01:22 Every single part of the area comes right up against that axis of revolution.

01:26 So this is a disk setup.

01:28 And for the disk setup, we know volume is pi times the integral of r squared...

Please give Ace some feedback