Washer Method and Shell Method setup issues

[jonwill]

Question)
A solid is formed by revolving the area inside x^(2/3)+y^(2/3)=a^(2/3), a>0; to the right of x=0 and above y=0 about the line x=a. Find the Volume of the solid using;
A) The washer Method
B) Cylindrical Shell Method

I am at a loss at how to set up this problem...I understand how to do the two methods once this problem is set up but I am just unable to do so.

I was able to solve for a and got a=(x^(2/3)+y^(2/3))^(3/2), but from there I am lost. I am new to the forum and only wished I had found it sooner. Everyone here seems very bright and knowledgeable.

Thanks for your time,
Jon

 

[tiny-tim]

Welcome to PF!

Hi Jon! Welcome to PF!

(try using the X² icon just above the Reply box )

Why are you solving for a? a is a constant

Both the washer method and the cylindrical shell method involve slicing the volume into easy-to-calculate regions.

The washer method uses "horizontal" slices, each has height dy and is shaped like a washer, ie it has an inner radius and an outer radius, so find the radii of each slice as a function of y, then find the volume (it will be a multiple of dy, the height), and integrate.

The cylindrical shell method uses "cookie-cutter" slices, each has thickness dx and is shaped like the round skin of a cylinder, so find the height of each slice as a function of x, then find the volume (it will be a multiple of dx, the thickness), and integrate.

What do you get? ​

 

[jonwill]

Ok Thank you,
Solving for y:
y=(a^2/3-x^2/3)^3/2

How would I plug this into the formula for the washer method? I get very confused when I am only given an "a" constant to work with instead of a solid number.

V=\Pi\int^b_a((R(x))²-(r(x))²)dx

 

[tiny-tim]

Hi Jon!

(have a pi: π and an integral: ∫ )

Ok Thank you,
Solving for y:
y=(a^2/3-x^2/3)^3/2

How would I plug this into the formula for the washer method? I get very confused when I am only given an "a" constant to work with instead of a solid number.

V=\Pi\int^b_a((R(x))²-(r(x))²)dx

Isn't the outer radius always "a"?

And what are your limits "a" and "b" supposed to be?

And i see you're using dx …

what slice is dx the width of?​