Question

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

Answer 1

Answer: V = $\frac{1024}{15}\pi$

Step-by-step explanation: The graph for function $y=8-8x^{2}$ 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.:

$V=\pi\int\limits^a_b {[f(x)]^{2}} \, dx$

where

f(x) is radius

dx is representative of height.

For the function above, volume is:

$V=\pi\int\limits^a_b {[(8-8x^{2})]^{2}} \, dx$

$V=\pi\int\limits^a_b {8^{2}(1-x^{2})^{2}} \, dx$

$V=64\pi\int\limits^a_b {(1-2x^{2}+x^{4})} \, dx$

$V=64.\pi[x-\frac{2}{3}x^{3}+\frac{x^{5}}{5} ]$

Using limits -1 to 1:

$V=64\pi(\frac{16}{15} )$

$V=\frac{1024}{15}\pi$

Volume of the rotated solid created by $y=8-8x^{2}$ about the x-axis is $\frac{1024}{15}\pi$

cubic units.

Answer 2

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:

$V = \dfrac{1024\pi}{15} \: \rm unit^3$

How to get the volume for  solid of revolution of given curve?

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:

$V = \int_{-1}^{+1} \pi y^2 dx = \pi\int_{-1}^{1} (8-8x^2)^2dx\\\\V = 64\pi\int_{-1}^{1} (1-x^2)^2dx = 84\pi\int_{-1}^{1} (1+ x^4 - 2x^2)dx \\\\V = 64\pi[x+ x^5/5 - 2x^3/3]_{x=-1}^{x=1} \\\\V = 128\pi \times \dfrac{8}{15} = \dfrac{1024\pi}{15} \: \rm unit^3$

Thus,

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

$V = \dfrac{1024\pi}{15} \: \rm unit^3$

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