Cylindrical shells (parts in details)?

  1. #f(x)=x^2#, #y=0#, #x=1#
    Set up a volume (shell) derivative based on the following axes of rotation:
    (A) around #x=-2#
    (B) around #x=2#
    (C) around #y=-2#
    (D) around #y=2#

  2. Can you use washers for (A) and (B) in the previous question? Why or why not?

  3. "Pikachu claims that no matter what kind of problem, you can always use disks/washers and shells. Is this true? Explain."

Answer 1

I have some parts of this question answered; feel free to check/change where needed

Here are my answers for number 1 #a=0# and #b=1#; (A) #\int_0^1[2\pi(x-(-2))(x^2)]dx=2\pi\int_0^1[(x+2)(x^2)]dx#
(B) #\int_0^1[2\pi(2-x)(x^2)]dx=2\pi\int_0^1[(2-x)(x^2)]dx#
(C) #\int_0^1[2\pi(y-(-2))(1-\sqrt{y})]dy=2\pi\int_0^1[(y+2)(1-\sqrt{y})]dy#
(D) #\int_0^1[2\pi(2-y)(1-\sqrt{y})]dy=2\pi\int_0^1[(2-y)(1-y)]dy#
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Answer 2

For attempted answers to questions 2 and 3. please see below.

Washers: #V = pi int_0^1 (9-(sqrty+2)^2) dy = (11pi)/6#
(B) Shells: #int_0^1 (2-x)x^2 dx = 5/6pi#
Washers: #pi int_0^1 ((2-sqrty)^2 - 1)dy = 5/6pi#

Obviously Pikachu is not correct. You cannot use discs/washers or shells to solve a related rates problem.

In theory, for any solid of revolution, we can use either. But consider the following problem.

Find the volume that results if the region bounded by the curve #y = xsinx# and the #x#-axis from the origin to #x=pi# is rotated about the #y#-axis..

graph{xsinx [-1.456, 4.702, -0.76, 2.318]}

Using shells, this is #pi int_0^pi x^2sinx dx = pi^2 -4#
To use washers we need expressions for the #x# values on the left and right in terms of #y#.
Since #y = xsinx# cannot be solved using standard mathematical functions, our best hope would probably be to find a series approximation to the inverse function. (I think that one exists.) So we could do it using washers (maybe), but we sure wouldn't want to.
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Answer 3

Cylindrical shells refer to hollow three-dimensional shapes formed by rolling up a flat surface, such as a rectangle or a circular segment, into a cylinder. They are commonly used in geometry and calculus to calculate volumes of complex shapes. The formula for finding the volume of a cylindrical shell is V = 2πrh, where V is the volume, r is the radius of the shell, and h is the height of the shell. This formula accounts for the surface area of the outer and inner surfaces of the shell. Cylindrical shells are often employed in problems involving solids of revolution, where a two-dimensional shape is rotated around an axis to form a three-dimensional object. They are also used in engineering and architecture for designing and analyzing structures like pipes, tanks, and columns.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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