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  

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