the answer should involve fracions and pi Region R is the region bounded on the

Question

the answer should involve fracions and pi

Region R is the region bounded on the left by the y-axis and on the right by the graph of X = 1 - y^2. Use the Method of Cylindrical Shells to find the volume of the solid of revolution generated by revolving region R about the line y = -5. (Enter a mathematical expression.) [40/6 pi] x Recall the Method of Cylindrical Shells. The volume of the solid generated by revolving the region enclosed by the curves x = f (y) and x = g (y) from y = a to y = b (where f (y) > g (y) for all y in the interval [a, b]) about the horizontal line y = c (where either c = b) is given by V = 2pi integral a to b |y - c|(f (y) - g (y)) dy. What does a sketch of the described region look like? How are the limits of integration found?

Explanation / Answer

given functions: x=1-y2 and y=-5

                     y2=1-x and y=-5

                    y=sqrt(1-x) and y=-5

height ----> -5- (sqrt(1-x)
radius ====> x
limits:
when y = -5
-5 = sqrt(1-x)
(-5)2=1-x
x=1-(-5)2
x= -24 &,26
volume:V=-24 to 26 r.h dx
            =-24 to 26 x.(-5-sqrt(1-x) dx

      this is procedure to get answer 40 pi/6

           =40 pi/6

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