How do you find the volume of the solid generated by revolving the region bounded by the curves y = x^(1/2), y = 2, and x = 0 rotated about the x=-1?

Answer 1

#V=(176pi)/15~=36.86#[cubic units]

Refer to the figure below

The generator region is Region A limited by #y=x^(1/2)#, #y=2# and #x=0# (the y-axis).

Since the revolution is around an axis parallel to the y-axis, it's convenient to use the inverse function
#f(y)=y^2#

We can use a trick to solve this problem: translating the y-axis to the position of the axis of the revolution. Consequently
#y=(X-1)^(1/2)#
#g(y)=y^2+1#

We only have to remember to exclude Region B, since now the old #x=0# became #x=1#

Applying the formula for volume of a solid of revolution (with a hole), with axis of revolution in y-axis:

#V=pi int_a^b [R(y)^2-r(y)^2]*dy#, where #R(y)# is the external radius and #r(y)# is the internal radius
#V=pi int_0^2 [(y^2+1)^2-1^2]*dy#
#V=pi int_0^2 (y^4+2y^2+cancel1-cancel1)*dy#
#V=pi int_0^2 (y^4+2y^2)*dy#
#V=pi (y^5/5+2/3y^2)|_0^2#
#V=pi(32/5+2/3*8)=pi(96+80)/15=(176pi)/15~=36.86# [cubic units]

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Answer 2

To find the volume of the solid generated by revolving the region bounded by the curves ( y = \sqrt{x} ), ( y = 2 ), and ( x = 0 ) rotated about the line ( x = -1 ), you can use the method of cylindrical shells. The integral setup for this problem would be:

[ V = \int_{0}^{4} 2\pi \left( x + 1 \right) \left( 2 - \sqrt{x} \right) , dx ]

Where:

  • ( V ) represents the volume of the solid.
  • The limits of integration ( 0 ) and ( 4 ) are determined by the intersection points of the curves ( y = \sqrt{x} ) and ( y = 2 ).
  • ( 2\pi ) is a constant factor that appears when calculating volumes of solids of revolution.
  • ( x + 1 ) represents the radius of the cylindrical shell.
  • ( 2 - \sqrt{x} ) represents the height of the cylindrical shell.

You would then integrate this expression with respect to ( x ) to find the volume of the solid.

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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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