# How do you find the volume bounded by #x^2y^2+16y^2=6# and the x & y axes, the line x=4 revolved about the x-axis?

= 3.7011 cubic units, nearly.

The volume is

= 3.7011 cubic units, nearly.-

graph{x^2y^2+16y^2-6=0 [-10, 10, -5, 5]}

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To find the volume bounded by ( x^2y^2 + 16y^2 = 6 ) and the x and y axes, with the line ( x = 4 ) revolved about the x-axis, you can use the method of cylindrical shells.

The formula for the volume of a solid generated by rotating a curve ( y = f(x) ) from ( x = a ) to ( x = b ) about the x-axis using cylindrical shells is:

[ V = \int_{a}^{b} 2\pi x \cdot f(x) , dx ]

In this case, we need to express ( y ) in terms of ( x ) from the given equation ( x^2y^2 + 16y^2 = 6 ), which simplifies to ( y = \frac{\sqrt{6}}{\sqrt{x^2 + 16}} ).

The bounds of integration ( a ) and ( b ) can be determined by finding the x-intercepts of the curve, which are the points where it intersects the x-axis. Solving ( x^2y^2 + 16y^2 = 6 ) when ( y = 0 ) gives us ( x^2 = 6 ), so ( x = \pm \sqrt{6} ). We'll integrate from ( x = -\sqrt{6} ) to ( x = 4 ).

Now, we'll plug ( f(x) = \frac{\sqrt{6}}{\sqrt{x^2 + 16}} ) into the formula and solve the integral:

[ V = \int_{-\sqrt{6}}^{4} 2\pi x \cdot \frac{\sqrt{6}}{\sqrt{x^2 + 16}} , dx ]

This integral might be challenging to solve directly. However, recognizing that the integrand is even, we can integrate from ( x = 0 ) to ( x = 4 ) and double the result:

[ V = 2 \int_{0}^{4} 2\pi x \cdot \frac{\sqrt{6}}{\sqrt{x^2 + 16}} , dx ]

[ = 4\pi \int_{0}^{4} \frac{x\sqrt{6}}{\sqrt{x^2 + 16}} , dx ]

This integral might require a trigonometric substitution. Once solved, it will give you the volume of the solid of revolution.

Therefore, the volume bounded by ( x^2y^2 + 16y^2 = 6 ) and the x and y axes, with the line ( x = 4 ) revolved about the x-axis, can be found using the cylindrical shells method and the definite integral as described above.

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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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