How do you find the volume of the solid formed by rotating the region enclosed by ?
about the xaxis.
about the xaxis.
Awesome Ken...
Here is a Maple rendering of the rotation!
By the way, you have the coolest name in the world.
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0.9117
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We'll use a method call the disk method to do this.
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To find the volume of the solid formed by rotating a region enclosed by a curve about an axis, you can use the method of cylindrical shells or disk/washer method depending on the shape of the region and the axis of rotation.

Cylindrical Shells Method:
 For regions enclosed by curves that are parallel to the axis of rotation, you can use the cylindrical shells method.
 The formula for the volume of the solid generated by cylindrical shells is given by: [ V = \int_{a}^{b} 2\pi x \cdot f(x) , dx ]
 Where ( f(x) ) represents the height of the shell at position ( x ), and ( a ) and ( b ) are the limits of integration.

Disk/Washer Method:
 For regions enclosed by curves that are perpendicular to the axis of rotation, you can use the disk or washer method.
 The formula for the volume of the solid generated by disks or washers is given by: [ V = \pi \int_{a}^{b} [f(x)]^2 , dx ]
 Where ( f(x) ) represents the distance from the curve to the axis of rotation at position ( x ), and ( a ) and ( b ) are the limits of integration.
To use these methods, you need to:
 Identify the axis of rotation.
 Determine whether the curves enclosing the region are parallel or perpendicular to the axis of rotation.
 Choose the appropriate method accordingly.
 Set up the integral based on the method chosen.
 Evaluate the integral to find the volume.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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