Use the disk or washer method to find the volume of the...

Use the disk or washer method to find the volume of the solid of revolution that is formed by revolving the region bounded by the graphs of the given equations about the indicated line or axis.
$y=x^{3}-x, y=0 ; \quad x$ -axis

Key Concepts

Washer Method

The washer method is an extension of the disk method used when the solid of revolution has a hollow region (a hole). In this case, each cross-sectional slice is a washer, which is a disk with a hole. The volume is computed by subtracting the inner radius area from the outer radius area for each slice and integrating these differences along the axis of rotation.

Integration Bounds

Determining integration bounds is essential for setting up the volume integral correctly. These bounds are the limits over which the region extends, typically found by identifying points of intersection between the curves defining the region. The correct bounds ensure that the volume calculation covers the entire region being rotated.

Solid of Revolution

A solid of revolution is a three-dimensional object obtained by rotating a two-dimensional region around a given axis. This concept is key in calculus for studying and computing volumes of objects with symmetry around an axis, often using integral calculus methods.

Disk Method

The disk method involves slicing the solid perpendicular to the axis of rotation to create disks (or circles) whose volumes can be computed. When the region being rotated does not have a gap, each thin slice is a disk with a radius defined by the distance from the axis of rotation to the curve, and the volume is found by integrating the area of these disks over the interval.

Transcript

00:01 So this graph of x cubed minus x looks something like this.

00:06 And you could double check where it crosses, if you said the cross of the x -axis, if you set it equal to zero.

00:13 You could factor out in x.

00:16 And then when you solve this, just remember that we could have this piece equal zero, or x could equal positive or negative 1, because it would be square negative 1 minus 1 is 0, and 0 times anything is 0.

00:30 So here's my confusion on this problem.

00:33 Do they want just this region rotated around the x -axis, or do they want both regions rotated around the x -axis? so what i'm going to do first is just do one of them.

00:48 So i'm just going to do the integral from 0 to 1.

00:50 It is the disk method, so just that function squared in terms of x.

00:59 And what i can do is just take the function.

01:02 Well, what i'm actually going to do because it's below the x -axis is i'm going to negate each piece, negative x cubed plus x.

01:09 That way i get a positive answer.

01:12 Now, if they want both regions in the second quadrant as well, then i could give you that answer is just going to be double because with an odd function and they're equal to each other.

01:24 So let's go ahead and do this.

01:26 So what i would do first is find the, like go ahead and square that.

01:31 It'll be x to the 6 power...

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