Disk method question

Question

Hi,

the question says:

find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x=5

given functions:

y=x, y=0, y=4, x=5

The problem I have with this question is finding the radius.

I know it is solved by using pi*integral(R^2)dy

but I don't understand how to find R. In the solution it says r=5-x and then since y=x it turns to 5-y which I understand because it rotates in a horizontal direction but I don't understand how the textbook got r=5-x.

please help

Answer 1

Since the line x = 5 is the center of rotation it is also the center of each circular disk that is formed by the rotation.  

 

5 - x is simply the horizontal distance between any point on the line y = x and the line x = 5 (taking into consideration the lower bound y = 0 and upper bound y = 4, which means that 0≤x≤4).  

 

It might be helpful to draw a graph of the lines y = x and x = 5 and then imagine what would happen if you rotated the line y = x out of the page and around the line x = 5.  

 

Note that the same process is used to find the radius when you rotate about the y-axis, but in that case the equation of the line of rotation is x = 0, so the radius of the circular disk is simply (x - 0) = x.