Find the volume of the solid generated by revolving the...

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the $x$ -axis.
$$y=\cos \frac{x}{2}, \quad y=\sin \frac{x}{2}, \quad x=0, \quad x=\frac{\pi}{2}$$

Key Concepts

Trigonometric Functions

When the bounding curves are given by trigonometric functions, it is important to understand their behavior and properties. Knowledge of sine and cosine functions, including their range, periodicity, and symmetry, helps in determining the correct radii and setting up the volume integral accurately.

Definite Integration

Definite integrals are used to accumulate infinitely small volume elements across the bounds of the region being revolved. They provide the mathematical framework for summing these elements to calculate the total volume, with the limits of integration determined by the boundaries of the region.

Determining Bounded Regions

Identifying the limits of integration and the boundaries of the region is a key step. This involves finding where the curves intersect with each other and with the given vertical or horizontal lines, which defines the interval over which to integrate.

Volume of Revolution

This concept involves creating a three-dimensional solid by rotating a two-dimensional region about an axis. In calculus, it is used to compute the volume of such solids by integrating cross-sectional areas over a given interval. Understanding this idea is crucial when the problem requires determining the total volume of a shape generated by rotation.

Washer Method

A specific technique for finding volumes of revolution when the solid has a hollow cavity, the washer method involves subtracting the volume of the inner solid from the outer solid. It uses the idea of slicing the region into washers (disks with holes) whose volume elements are computed by the difference of the squares of the outer and inner radii.

Transcript

00:02 Let me start off the problem by sketching the region.

00:05 The region here is the region bounded by the functions y equals cosine of x over 2, y equals sine of x over 2, x equals 0, and x equals pi over 2.

00:25 So let me just sketch the region on the right side here.

00:29 And i'll try to be cleared with the including the image of the revolution as well.

00:35 So first we have a sine of x and sign of x over 2.

00:41 It will look something like this.

00:43 I imagined, so y equals to sine of x over 2.

00:47 Then we also have cosine, cosine will look something.

00:51 Just exaggerate a little bit.

00:52 It's going to look something like this.

00:55 Let's call this y equals cosine of x over 2.

00:59 The region that we are referring to should be obvious.

01:02 It is the region bounded by all of them, although given information, which is that when we revolve this image about the x -axis, we're going to get a three -dimensional shape.

01:17 So i'm just going to visualize how this shape is going to look like.

01:22 I'm going to look like this and like that and like that.

01:28 Here i'm going to point out to two important things.

01:31 So here we're going to call this distance right here, little r -fx, and we're going to call this distance...

Please give Ace some feedback