Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. y = 8 − 8x2, y = 0

V =

Step-by-step explanation: The graph for function shows the maximum height is 8and x-axis limits are -1 and 1, which will also be the limits of the integration.

For a volume of a solid created by revolution of a single function, it is appropriate to use the Disk Method, i.e.:

where

f(x) is radius

dx is representative of height.

For the function above, volume is:

Using limits -1 to 1:

Volume of the rotated solid created by about the x-axis is

cubic units.

You can use the fact that when the given curve is revolved around x axis, each height y will contribute as radius at that point x and there will be a circle. Multiply that area of circle with dx to get infinitesimal volume and integrate it.
The volume of the solid obtained by revolving the given curve and bounded by y = 0 (x axis) is given by:

Since the solid will be bounded by y = 0, thus, the curve will be used from x = -1 to x = +1 (since outside that limit, the curve goes down the y =0(x axis) line.(see the graph attached).

Since at each input x in [-1,1], we have the output as radius of a vertical circle standing there with center at (x,0), thus, integrating the area times dx from x = -1 to x = +1, we get:

Thus,

The volume of the solid obtained by revolving the given curve and bounded by y = 0 (x axis) is given by:

Learn more about solid of revolution here:

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step-by-step explanation: