# Find the area by using disk method

• MHB
• jaychay
In summary, the conversation discusses solving for the area R and using integrals to find the volume of a rotated shape. The final step involves finding the radius of a disk when rotating R about the line y = 1.
jaychay

The problem is to solve for the area R.

I have tried to do it many times.

Volume V1 is on x - axis
Volume V2 is on y=1

$\displaystyle V_1 - V_2 = \pi \int_1^4 [f(x)]^2 \, dx - \pi \int_1^4 [f(x)-1]^2 \, dx = 4\pi$

$\displaystyle \int_1^4 [f(x)]^2 \, dx - \int_1^4 [f(x)]^2 - 2f(x) + 1 \, dx = 4$

$\displaystyle \cancel{\int_1^4 [f(x)]^2 \, dx} - \cancel{\int_1^4 [f(x)]^2 \, dx} + \int_1^4 2f(x) - 1 \, dx = 4$

note $\displaystyle \int_1^4 f(x) \, dx = R + 3$
can you finish?

skeeter said:
$\displaystyle V_1 - V_2 = \pi \int_1^4 [f(x)]^2 \, dx - \pi \int_1^4 [f(x)-1]^2 \, dx = 4\pi$

$\displaystyle \int_1^4 [f(x)]^2 \, dx - \int_1^4 [f(x)]^2 - 2f(x) + 1 \, dx = 4$

$\displaystyle \cancel{\int_1^4 [f(x)]^2 \, dx} - \cancel{\int_1^4 [f(x)]^2 \, dx} + \int_1^4 2f(x) - 1 \, dx = 4$

note $\displaystyle \int_1^4 f(x) \, dx = R + 3$
can you finish?

Can you explain to me please where did (f(x)-1)^2 come from ? and why you have to put -1 behind f(x) ?

jaychay said:
https://www.physicsforums.com/attachments/10787

Can you explain to me please where did (f(x)-1)^2 come from ? and why you have to put -1 behind f(x) ?

rotating R about the line y = 1 using disks ... what is the radius of a disk in this case?

## What is the disk method?

The disk method is a mathematical technique used to find the area of a solid of revolution. It involves dividing the solid into infinitely thin disks and summing their areas to find the total volume.

## When is the disk method used?

The disk method is used when finding the area of a solid that is created by rotating a curve around an axis. This could be a circle, parabola, or any other curve.

## What are the steps for using the disk method?

The steps for using the disk method are as follows:

• 1. Identify the axis of rotation and the curve to be rotated.
• 2. Determine the limits of integration for the integral.
• 3. Set up the integral using the formula for the area of a disk.
• 4. Evaluate the integral to find the total area.

## What are the assumptions made when using the disk method?

The disk method assumes that the solid of revolution is made up of infinitely thin disks, and that the disks are all parallel to the axis of rotation. It also assumes that the curve being rotated is continuous and does not intersect itself.

## What are some common mistakes when using the disk method?

Some common mistakes when using the disk method include forgetting to square the radius in the formula for the area of a disk, using the wrong limits of integration, and not taking into account any holes or gaps in the solid of revolution.

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