When the segment of the line y = 8 from x = 2 to x = 10 is rotated about the x-axis what is the shape of the solid formed and its surface area and volume.
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A line segment of y = 8 from x = 2 to x = 10 is rotated about the x-axis. The graph of the line segment is:
When this line segment is rotated about the x-axis, the shape of the solid formed is that of a cylinder. The radius of the cylinder is equal to 8 and the length of the cylinder is 10 - 2 = 8.
The volume of the cylinder with radius r and length l is `V = pi*r^2*l` . The surface area of a cylinder open at both ends is `2*pi*r*l` . If the cylinder is closed the surface area is `2*pi*r*l + 2*pi*r^2` .
Here, r = 8 and l = 8. The volume of the cylinder is `pi*8^2*8 = 512*pi` . The surface area of an open cylinder is `2*pi*8*8 = 128*pi` and the surface area of a closed cylinder is `2*pi*8*8 + 2*pi*8^2 = 4*pi*64 = 256*pi` .
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