Skip to main content
Search IntMath
Close

450+ Math Lessons written by Math Professors and Teachers

5 Million+ Students Helped Each Year

1200+ Articles Written by Math Educators and Enthusiasts

Simplifying and Teaching Math for Over 23 Years

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.

IntMath forum | Applications of Integration

Shell Method [Solved!]

My question

URL:Shell Method | Brilliant Math & Science Wiki

Under the section, "When to use the shell method", with the first example, should the disks (blue rectangles) be horizontal rectangles located between the graph and line x = 1?

Relevant page

Shell Method | Brilliant Math & Science Wiki

What I've done so far

Not clear as to how they determined the region to subtract is the region bounded by y = 1,
y = `sqrt x`, and the y-axis.

X

URL:<a href="https://brilliant.org/wiki/shell-method/">Shell Method | Brilliant Math &amp; Science Wiki</a>

Under the section, "When to use the shell method", with the first example, should the disks (blue rectangles) be horizontal rectangles located between the graph and line x = 1?
Relevant page

<a href="https://brilliant.org/wiki/shell-method/">Shell Method | Brilliant Math &amp; Science Wiki</a>

What I've done so far

Not clear as to how they determined the region to subtract is the region bounded by y = 1,
y = `sqrt x`, and the y-axis.

Re: Shell Method

What they've done in their example is to find the volume of the cylinder generated when rotating the unit square (with vertices `[0,0]`, `[1,0]`, `[1,1]` and `[0,1]`) around the `y`-axis, then they've subtracted the volume of the cone-shaped part from that, to give the required funnel-shaped volume.

So their blue rectangles are correct how they have them (although I'd normally only have one rectangle, as a "typical" rectangle.)

We could have found that volume if the blue rectangle(s) is(are) where you thought it(they) should be (forming the funnel-shaped volume).

In that case, we have to proceed as follows:

For the function we need to integrate, we take the outer border (which is `x=1`, the line parallel to the `y`-axis) minus the inner border (the curve `y=sqrt(x)` that is, `x=y^2`)

Volume `= pi int_0^1 (1-(x)^2)dy`

`= pi int_0^1 (1-y^4)dy`

`= pi [y-y^5/5]_0^1`

`= pi[(1-1/5)-(0-0)]`

`= (4pi)/5`

It amounts to the same thing as their answer, just that it's written in one expression.

X

What they've done in their example is to find the volume of the cylinder generated when rotating the unit square (with vertices `[0,0]`, `[1,0]`, `[1,1]` and `[0,1]`) around the `y`-axis, then they've subtracted the volume of the cone-shaped part from that, to give the required funnel-shaped volume.

So their blue rectangles are correct how they have them (although I'd normally only have one rectangle, as a "typical" rectangle.)

We could have found that volume if the blue rectangle(s) is(are) where you thought it(they) should be (forming the funnel-shaped volume).

In that case, we have to proceed as follows:

For the function we need to integrate, we take the outer border (which is `x=1`, the line parallel to the `y`-axis) minus the inner border (the curve `y=sqrt(x)` that is, `x=y^2`)

Volume `= pi int_0^1 (1-(x)^2)dy`

`= pi int_0^1 (1-y^4)dy`

`= pi [y-y^5/5]_0^1`

`= pi[(1-1/5)-(0-0)]`

`= (4pi)/5`

It amounts to the same thing as their answer, just that it's written in one expression.

Re: Shell Method

Excellent! I have the same question as you. Now, I can solve these questions thanks to your help.
color tunnel

X

Excellent! I have the same question as you. Now, I can solve these questions thanks to your help.
 <a href="https://color-tunnel.com">color tunnel</a>

Reply

You need to be logged in to reply.

Related Applications of Integration questions

Applications of Integration lessons on IntMath

top

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.