The disc method allows us to find the volume of a solid in a 2 dimensional space by taking the cross section of a solid.  When taking the cross section, you are slicing perpendicular to the axis of rotation (usually either x or y axis).  Then we sum infinitely many of these cross sections to obtain a value for the volume, sort of like integrating under the curve.

The disc method is based on the volume of a cylinder, .  In the formula for the disc method,  is the volume of the solid,  is the smallest value of  of ,  is the largest value of  of ,  is the radius of the disc, and  is the height of the disc.

First we must find the cross sectional area.  To do this, we need the distance from the x-axis to the function (the radius of each disc).  If we think about it, this is just the function itself.

Now we can plug into our disc method formula:  or .

Use a u-substitution where 

First set up the disc method formula then plug in the given function.

These two methods are used in different situations so their accuracy is not comparable.  The Washer Method is like using the disc method when considering the difference between two discs.  Their accuracy is similar and you should use the Washer method when you have an inner and outer radius and the disc method when you have only one radius.

The washer method uses an adaptive version of the volume of a cylinder.  If we think of a washer, it is disc-like but has a hole in the middle.  So we need to consider both an inner and outer radius.  Similar to the disc method we are taking cross sectionals of a solid but this solid has a hole in the middle.  We sum infinitely many of these cross sections to obtain a value for the volume.

The washer method uses an adaptive version of the volume of a cylinder.  If we think of a washer, it is disc-like but has a hole in the middle.  So we need to consider both an inner and outer radius.  For this reason, you must consider two independent functions as two separate radii measured as the distance from the axis of rotation.

When trying to think about our integration limits remember that  this function starts at the origin and moves in the positive  and  direction, so our lower limit will be .  When thinking about our upper limit we must find when these two functions intersect.  To do so, we set them equal to one another:

So our upper limit is .  Next we must think about the volume of the solid we are trying to find.  We will have two radii to consider since there are two different  equations.  Let’s rearrange both in terms of .

The function that would be ‘on top’ in a graph would be , so this is our inner radius ( ) making our outer radius ( ).

Now we can plug into our washer method formula 

Explanation: First we must try to find our integration limits.  So we need to find where these two functions intersect.  We do so by setting them equal to one another.

So  is one limit.  If we look at both functions we see they both pass through the origin, so  is our lower limit and  is our upper limit.  Next we must think about the volume of the solid we are trying to find.  We will have two radii to consider since there are two different  equations.  Let’s rearrange both in terms of .

The function that would be ‘on top’ in a graph would be .  So this is our inner radius ( ) making  our outer radius( ).

Now we can plug into our washer method formula 

Now we are considering a solid who is not rotating around the x or y-axis but the line .  First we must find the limits of integration.  We can look at their points of intersection to find the upper and lower limits.  We do this by setting them equal to one another.  

So our lower limit is  and the upper limit is .

Now we must determine our radii.  Our inner radius ( ) will be :

For the outer radius ( ):

Now we can plug this into our washer method formula