If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Integral Calculus

### Unit 3: Lesson 10

Volume: disc method (revolving around other axes)

# Disc method rotating around vertical line

Volume of solid created by rotating around vertical line that is not the y-axis using the disc method. Created by Sal Khan.

## Want to join the conversation?

• What is a 'principle root?' Is that just another name for a square root? •   Principle root is the positive square root. For example, √25 asks the question "what number when multiplied together gives 25?" This question has two answers: 5 and -5. The principle root of 25 however is unambiguously 5. Unless you see ± before the radical symbol, it usually means the principle root.
• The diagram is confusing to me because the parabola at -2 would be at y = +3 and revolve from the left piece to a cup or from the right piece to an inverted rim. This diagram looks like the parabola was transformed so that it minimum occurs at -2,-1. • You aren't the first person to find this example confusing, and I suspect Sal would spend a little more time explaining the setup, or maybe change his diagram a bit, if he were to redo it. The main thing you need to understand for proper visualization is that we/re using only the part of the parabola from x = 0 to x = 2 (which translates into the right branch of the parabola from y = -1 to y = 3).

If that clue isn't enough to give you a clear visualization, step away from this example for a minute and imagine a different one, where we have the line segment traced by x = 2 from y = -1 to y = 3, and we rotate it around the vertical line at x = -2. This would simply be a cylinder centered at x = -2. Now imagine that we change the vertical line segment that we rotated by bending in the bottom of it, not all the way to the center of the cylinder (at x = -2) but only to x = 0, and we rotate that curve around the same vertical at x = -2. The bottom of the figure would still have a flat region, just as the cylinder did, but now the flat region is smaller because its radius extends only from x = -2 to x = 0. That's the "gumdrop-shaped" figure Sal drew in his diagram.

To give it 3 dimensions he drew it at an angle instead of a straight-on cross section. Drawing it this way called for the bottom of the figure to be shown as curved instead of straight as it would be in a cross section, and that makes it look as if the parabola continues downward to the left from x = 0, when in reality the parabola ends at x = 0 and the curved line to the left of that point is a perspective view of the round, but flat, bottom of this figure.
• Why is this evaluated using the disk method and not the washer method? If you're evaluating it from -1 to 3 on the y-axis, wouldn't that leave a hollowed out section in the middle around the axis of rotation? • considering y=x^2 - 1 is a parabola with a local minimum at x = -1, wouldn't the solid created by rotating around x=-2 require the shell method? • • Hi!

We rotated `y = x^2-1` around the line `x = -2`. `x = -2` is parallel to our y-axis. Therefore, Sal picked 2 points on the y-axis as our interval (`y = -1`, `y = 3`).

Think about it, the sum of all the discs that are being created when we let `y = x^2-1` rotate around the line `x = -2` is the total volume of the figure created when `y = x^2-1` rotated around `x = -2`. This is basically the same thing as letting `y = x^2-1` rotate around the y-axis but instead, we rotated it around a line that is parallel to it, in this case `x = -2`. The difference is that the radius becomes different but the same sort of discs are created inside of the figure.

To answer your question: By picking `-1` and `3` we will calculate the volumes of all the discs between `y = -1` and `y = 3`, which gives us the volume of the figure between `y = -1` and `y = 3`.

Hope I could be of any help!

// Kris
(1 vote)
• Why use a principle root only and not include the negative root? • The equation y=x^2-1 is that of a shifted parabola with point of contact with the line x= (-2) at (-2,3). Then on rotating the figure around the axis x= -2 shouldn't we get a cup like figure ( as shown in the video) along with a conical figure in it ( not shown in the video ) ? • Earlier, you subtracted a function by a constant (x=2 or something) but now, you are subtracting a constant (in this case -2) from your function. How do you know if the given function should be subtracted by the given constant or vice versa?
(1 vote) • Based on how the area is between y = x^2 - 1 and the y axis, and the rotation is around x =-2, shouldn't there be a hole in the middle of the gumdrop shape?  