How do you know when to use the shell method or the disk method?
You'll get the same answer either way.
Sometimes one leads to an integral that a particular person finds easier to evaluate, but what is easier varies between people.
I have a preference for doing a single integral.
Again, that is my preference. As a student, perhaps because we learned disks/washers first, I preferred them.
In the end it doesn't matter. If you set it up one way and don't care for the looks of the integral, try setting it up the other way.
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You can determine whether to use the shell method or the disk method to find the volume of a solid of revolution by considering the axis of rotation and the orientation of the slices.

Shell Method:
 Use when the axis of rotation is horizontal or vertical (parallel to the slices).
 Slice the region perpendicular to the axis of rotation.
 Each slice is a cylindrical shell (a hollow cylinder).
 Integrate along the axis of rotation (usually (x) or (y)).

Disk Method:
 Use when the axis of rotation is perpendicular to the slices.
 Slice the region parallel to the axis of rotation.
 Each slice is a disk (or washer if there's a hole).
 Integrate perpendicular to the axis of rotation (usually (x) or (y)).
To decide which method to use:
 Visualize the region and the axis of rotation.
 Determine the orientation of the slices needed for each method.
 Choose the method that aligns with the orientation of the slices and the axis of rotation.
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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.
 Find the solution of the differential equation # x(1+y^2)dxy(1+x^2)dy =0#?
 What is the surface area produced by rotating #f(x)=sinx/cosx, x in [0,pi/4]# around the xaxis?
 How do you find the volume of the solid generated by revolving the region bounded by the graphs #y=3/(x+1), y=0, x=0, x=8#, about the x axis?
 How do you find the distance travelled from t=0 to #t=2pi# by an object whose motion is #x=cos^2t, y=sin^2t#?
 What is the surface area of the solid created by revolving #f(x) = lnx , x in [2,3]# around the x axis?
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