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^2+1)#, x=0, x=1, y=0 revolved about the y-axis?

Answer 1

Volume #= piln2 #

If you imagine an almost infinitesimally thin vertical line of thickness #deltax# between the #x#-axis and the curve at some particular #x#-coordinate it would have an area:

#delta A ~~"width" xx "height" = ydeltax = f(x)deltax#

If we then rotated this infinitesimally thin vertical line about #Oy# then we would get an infinitesimally thin cylinder (imagine a cross section through a tin can), then its volume #delta V# would be given by:

#delta V~~ 2pi xx "radius" xx "thickness" = 2pixdeltaA=2pixf(x)deltax#

If we add up all these infinitesimally thin cylinders then we would get the precise total volume #V# given by:

# V=int_(x=a)^(x=b)2pixf(x) dx #

So for this problem we have:

# V = int_0^1 2pix 1/(x^2+1) dx #
# \ \ \ = pi int_0^1 (2x)/(x^2+1) dx #
# \ \ \ = pi ln|x^2+1)|_0^1 #
# \ \ \ = pi (ln2-ln1)#
# \ \ \ = piln2#

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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^2 + 1} ), ( x = 0 ), ( x = 1 ), and ( y = 0 ) revolved about the y-axis, follow these steps:

  1. Determine the limits of integration. In this case, since the region is bounded by ( x = 0 ) and ( x = 1 ), the limits of integration for ( x ) will be from ( 0 ) to ( 1 ).

  2. Identify the height of each cylindrical shell. The height of each shell will be the function ( y = \frac{1}{x^2 + 1} ).

  3. Determine the radius of each cylindrical shell. The radius will be the distance from the axis of rotation (y-axis) to the function ( y = \frac{1}{x^2 + 1} ) at each value of ( x ). Since we are revolving around the y-axis, the radius will be ( x ).

  4. Write the formula for the volume of each cylindrical shell. The volume ( V ) of a cylindrical shell is given by: [ V = 2\pi rh ] where ( r ) is the radius and ( h ) is the height.

  5. Integrate the expression for the volume of each cylindrical shell with respect to ( x ) over the given limits of integration (from ( 0 ) to ( 1 )). This will give the total volume of the solid of revolution.

[ V = \int_0^1 2\pi x \cdot \frac{1}{x^2 + 1} , dx ]

  1. Evaluate the integral to find the volume.

[ V = 2\pi \int_0^1 \frac{x}{x^2 + 1} , dx ]

You can solve this integral using methods such as substitution or partial fractions.

By following these steps, you can use the method of cylindrical shells to find the volume of the solid obtained by rotating the given region about the y-axis.

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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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