Finding the volume of a solid enclosed by two cylinders and planesConsider the solid E1 that is enclosed by the planes z = 0 and z = 5 and by the cylinders X 2 + y 2 = 9 and x 2 + y 2 = 16. Find the volume of E1. How to find the volume using two cylinders

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Finding the volume of a solid enclosed by two cylinders and planesConsider the solid E1 that is enclosed by the planes   z  =  0 and   z  =  5 and by the cylinders       X    2    +      y    2    =  9 and       x    2    +      y    2    =  16. Find the volume of E1. How to find the volume using two cylinders

Rylan White · Accepted Answer

Step 1There are two ways of doing it:1. Using formulas;2. Integrating.Let&#039;s use the formula that gives us the volume of a cylinder:       π    (  r  a  d  i  u  s      )    2    h  e  i  g  h  t. We are just gonna subtract the tinner from the bigger:       π    (  4      )    2    5  −      π    (  3      )    2    5  =  35      π  .Now let&#039;s integrate!:  ∭      E    1      d  V  =  V  o  l  (  E  1  )n order to have an easier integral, we should use cylindrical coordinates: (x,y,z) will &quot;become&quot;   (  ϑ  ,  r  a  d  i  u  s  ,  z  ). Therefore, (not forgetting the distortion):      ∫    0    5        ∫    0          2      π            ∫    3    4    (  r  a  d  i  u  s  )  d  r  d  ϑ  d  z  =  V  o  l  (  E  1  )Using Fubini&#039;s theorem:      ∫    0    5    d  z      ∫    0          2      π        d  ϑ      ∫    3    4    (  r  a  d  i  u  s  )  d  r  =  V  o  l  (  E  1  )Step 2And finally:  V  o  l  (  E  1  )  =  35  πFor solving such problems I have set a couple of steps that I encourage you following:- Draw the region,- Choose the coordinate system that fits better for the ocasion,- Set the limits of the variables,- And then integrate.It is important to explane why I chose those limits in respect to   r  ,  ϑ  ,  z. The last one is the easiest: the region is confined within the two planes,   z  =  0  ,  z  =  5; about the others: due to the symmetry of E1, you are going to rotate   2  π radians a segment of line the starts at 3 (the radius of the tinner cylinder) and ends at 4 (the radius of the bigger one).

Finding the volume of a solid enclosed by two cylinders and planes. Consider the solid E1 that is enclosed by the planes z = 0 and z = 5 and by the cylinders X^2 + y^2 = 9 and x^2 + y^2 = 16. Find the volume of E1.

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