Volume Solid of Revolution Shell Method

[efnoer]

How do we find the volume between the functions y=x^2-4x+5, y=3/2x, x=2, x=4 about the horizontal lines y=0, y=10, and y=-2? Please help!

 

[Deleted member 4993]

How do we find the volume between the functions y=x^2-4x+5, y=3/2x, x=2, x=4 about the horizontal lines y=0, y=10, and y=-2? Please help!

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[efnoer]

So far, we have given the horizontal lines the parameter (a). When the horizontal line would less than two, for example around the line x=-2 we go the integral of 2pi * (x-(-2))(-x^2+11/2x-5) dx, with the bounds 2 and 4. When the horizontal line is greater than 4, such as 10, it would be integral of 2pi*(10-x)(-x^2+11/2x-5) dx. Are these integrals correct? What should we do going forward?

 

[skeeter]

rotation of a region about a given horizontal line using the method of washers is [MATH]V = \pi \int_a^b [R(x)]^2 - [r(x)]^2 \, dx[/MATH], where [MATH]R[/MATH] is the farthest vertical distance between one of the functions and the axis of rotation, and [MATH]r[/MATH] is the nearer vertical distance between the other function and the axis of rotation


rotation about y = 0 (the x-axis) ...

[MATH]R(x) = \frac{3x}{2}[/MATH][MATH]r(x) = x^2-4x+5[/MATH]
rotation about y = 10 ...

[MATH]R(x) =10 - (x^2-4x+5)[/MATH][MATH]r(x) = 10 - \frac{3x}{2}[/MATH]
rotation about y = -2 ...

[MATH]R(x) = \frac{3x}{2} - (-2)[/MATH][MATH]r(x) = (x^2-4x+5) - (-2)[/MATH]

 

[efnoer]

@skeeter Thank you. We have solved these using washer methods. However, we need to use shell method to rotate about y-axis. Our original message had an error. It is x=0, x=10, x=-2. Thank you for your help.

 

[efnoer]

The shape in the graph is slightly complicated, so we were not sure how to find the integral when revolving over y-axis and vertical lines.

 

[Deleted member 4993]

@skeeter Thank you. We have solved these using washer methods. However, we need to use shell method to rotate about y-axis. Our original message had an error. It is x=0, x=10, x=-2. Thank you for your help.

Please show us the limits (for integration) for washer method.

 

[efnoer]

The bounds for the integration are a=2 b=4

 

[skeeter]

shells formed by rotating a vertical strip of area about a vertical line ...

[MATH]V = 2\pi \int_c^d r(x) \cdot h(x) \, dx[/MATH]
radius of rotation for the cylindrical shell,

[MATH]r(x) = x - \text{ (axis of rotation) if }x > \text{ (axis of rotation)}[/math][math] r(x) = \text{ (axis of rotation) } - x \text{ if } x < \text{ axis of rotation}[/MATH]
height of the cylindrical shell, [MATH]h(x) = \text{ upper function - lower function}[/MATH]

 

[efnoer]

@skeeter Wouldnt the shape have to be divided into three pieces and then integrated?

 

[efnoer]

nce at y=5 y=3

 

[skeeter]

no, just one integral

for rotation about x = 0 ...

[MATH]V = 2\pi \int_2^4 x\left[\frac{3x}{2} - (x^2-4x+5)\right] \, dx[/MATH]
the only change for the other two axes of rotation will be [MATH]r(x)[/MATH]
continue ...