How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y = 1/x^4#, y = 0, x = 1, x = 4 revolved about the x=-4?

Answer 1

#(57pi)/16#

the formula for the shell method is #int_a^b2pirhdx#
#a# and #b# are the x-bounds, which are x=1 and x=4, so #a=1# and #b=4#.
#r# is the distance from a certain x-value in the interval #[1,4]# and the axis of rotation, which is x=-4. #r=x-(-4)=x+4#
#h# is the height of the cylinder at a certain x-value in the interval #[1,4]#, which is #1/x^4-0=1/x^4# (because #1/x^4# is always greater than #0# and h must be positive).
plugging it all in: volume #=int_1^4(2pi(x+4)(1/x^4))dx# you should get: #(57pi)/16#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by ( y = \frac{1}{x^4} ), ( y = 0 ), ( x = 1 ), and ( x = 4 ) revolved about the line ( x = -4 ), follow these steps:

  1. Determine Limits of Integration: The region of integration lies between ( x = 1 ) and ( x = 4 ).

  2. Setup the Integral: The volume ( V ) is given by the integral ( V = \int_{1}^{4} 2\pi rh , dx ), where:

    • ( r ) is the distance from the axis of rotation to the shell, which is ( 4 + x ) in this case.
    • ( h ) is the height of the shell, which is ( \frac{1}{x^4} ).
  3. Evaluate the Integral: Integrate ( 2\pi (4 + x) \frac{1}{x^4} , dx ) from ( x = 1 ) to ( x = 4 ).

  4. Calculate the Volume: Evaluate the definite integral obtained in the previous step to find the volume of the solid.

  5. Final Step: Make sure to express the volume in exact or approximate form, as required.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7