Finding the volume of a region rotated about the y-axis.I'm having trouble trying to find the volume of the region formed by y = x 2 − 7 x + 10 and y = x + 3 rotated about the y-axis. I was able to graph it, but I'm having difficulty when trying to come up with the integral for it.Any help showing me how to get the integral I should use would be a huge help.

Question

Finding the volume of a region rotated about the y-axis.I&#039;m having trouble trying to find the volume of the region formed by   y  =      x    2    −  7  x  +  10 and   y  =  x  +  3 rotated about the y-axis. I was able to graph it, but I&#039;m having difficulty when trying to come up with the integral for it.Any help showing me how to get the integral I should use would be a huge help.

Tolamaes04 · Accepted Answer

Step 1This one may be easier to handle using the Method of Cylindrical Shells. Setting   x  +  3  =      x    2    −  7  x  +  10 we find that the curves meet at   x  =  1 and   x  =  7. Over the interval [1,7], the line   y  =  x  +  3 is the upper curve. When we take a vertical strip of width &quot;dx&quot; from x to   x  +  d  x, and rotate it, we get a cylindrical shell of radius x and height   (  x  +  3  )  −  (      x    2    −  7  x  +  10  )  =  −      x    2    +  8  x  −  7.Step 2Thus the volume is       ∫    1    7    2  π  x  (  −      x    2    +  8  x  −  7  )    d  x  .For the integration, multiply out and integrate term by term.

Jaylene Hunter · Accepted Answer

Explanation:You must split the integration. for   y  ∈  [  −      9    4    ,  4  ] the radii of the inner and outer annuli are the two roots of       x    2    −  7  x  +  (  10  −  y  )  =  0, whereas for   y  ∈  [  4  ,  10  ] the outer radius is the larger root of that equation, but the inner radius is given by   y  −  3

Find the volume of the region formed by y=x^2-7x+10 and y=x+3 rotated about the y-axis. I was able to graph it, but I'm having difficulty when trying to come up with the integral for it.

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