How do you find the volume bounded by #y^2=x^3-3x^2+4# & the lines x=0, y=0 revolved about the x-axis?

Answer 1

#= 4 pi#

It's this

but bounded in the first quadrant

A small element of width #delta x# and reaching from x-axis to the curve will have height #y#

And so will have volume, when revolved about x-axis, of #delta V = pi y^2 delta x#

So we can say that

# V = pi int_0^2 y^2 dx#

#= pi int_0^2 x^3-3x^2+4 dx#

#= pi [ x^4/4-x^3+4x ]_0^2#

#= 4 pi#

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Answer 2

To find the volume bounded by the curve (y^2=x^3-3x^2+4), the x-axis, and the y-axis, when revolved about the x-axis, you can use the method of cylindrical shells. The volume can be calculated using the formula:

[ V = 2\pi \int_{a}^{b} x\cdot |y| , dx ]

where ( a ) and ( b ) are the x-values where the curve intersects the x-axis. In this case, ( a ) and ( b ) are the x-intercepts of the curve. To find these intercepts, set ( y = 0 ) and solve for ( x ) in the equation ( y^2 = x^3 - 3x^2 + 4 ).

Once you find ( a ) and ( b ), integrate ( x \cdot |y| ) with respect to ( x ) from ( a ) to ( b ) and multiply by ( 2\pi ).

[ V = 2\pi \int_{a}^{b} x\cdot |y| , dx ]

[ V = 2\pi \int_{a}^{b} x \sqrt{x^3-3x^2+4} , dx ]

Evaluate this integral to find the volume.

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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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