Method of disk (in respect to y)

In summary, the problem is to find the volume of the solid of revolution obtained by rotating the region under the curve y = √x on the interval [0, 4] about the x-axis. The initial attempt to solve using the integral formula in terms of dx resulted in the correct answer of 8pi. However, upon deciding to use dy instead, it was noticed that the height and limits of integration remain the same. The correct integral to use is pi * (y^2)^2 from 0 to 2, giving a volume of 16/5 instead of 8pi. This is because the original function was y = √x, so when inverse it, we get x =
  • #1
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Homework Statement



Revolve the region under the curve y =√x on the interval [0, 4] about the x-axis and
find the volume of the resulting solid of revolution.

YET, NOW, I want to do in respect to y

Homework Equations


integral formula


The Attempt at a Solution



I did dx and I got 8pi which is corrected.

So I decided to do it with dy.

I looked at the graph and I saw the height would be the same (well because of the problem's nature). so a and b will stay the same

integral of pi*r^2
we know r is y^2 (since y^2 = x)
integral of pi * (y^2)^2 from 1 to 4

but I did not get 8pi
why?

thanks.
 
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  • #2
The integral you evaluated, gives the volume when the curve is rotated about the y-axis, not the x-axis.
 
  • #3
The original function was y = sqrt of x
so inverse it, we got x = y^2

you were right about the fact. i think the point of a and b should be 0 to 2, since y^2 = x, so (2^2) = 4

integral of pi * (y^2)^2 from 0 to 2
but i got 16/5 instead
 

What is the method of disk in respect to y?

The method of disk in respect to y is a mathematical technique used to calculate the volume of a solid of revolution by slicing the solid into thin disks parallel to the y-axis.

How is the method of disk in respect to y different from the method of disk in respect to x?

The method of disk in respect to y is similar to the method of disk in respect to x, but instead of slicing the solid parallel to the x-axis, it is sliced parallel to the y-axis. This is useful when the solid being rotated has a more complex shape along the y-axis.

What are the steps involved in using the method of disk in respect to y?

The steps involved in using the method of disk in respect to y are: 1) Identify the function that defines the shape of the solid along the y-axis, 2) Determine the limits of integration, or the range of y-values over which the solid is being rotated, 3) Set up the integral using the formula for the volume of a disk, and 4) Solve the integral to find the volume of the solid.

When should the method of disk in respect to y be used?

The method of disk in respect to y should be used when the solid being rotated has a more complex shape along the y-axis and the volume cannot be easily calculated using other methods, such as the method of disk in respect to x or the method of cylindrical shells.

What are some real-life applications of the method of disk in respect to y?

The method of disk in respect to y has many real-life applications, such as calculating the volume of objects like vases, bottles, and cans that have a more complex shape along the y-axis. It is also used in physics and engineering to calculate the moment of inertia of objects, which is important in understanding their rotational motion.

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