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The volume will be 3π + π²/16 if the solid obtained by rotating the region bounded by the given curves about the specified axis. y=0, y= cos(4 x), x = π/8, x = 0 about the axis y= -4f
It is defined as the mathematical calculation by which we can sum up all the smaller parts into a unit.
We have curves,
[tex]\rm y=0, y= \cos(4 x), x = \frac{\pi}{8}, x = 0[/tex]
Here the axis is y = -4f, but not mentioned the value of f.
We are assuming the f = 3/2
So the axis will be y = -4(3/2) = -6
Inner radius = 6
Outer radius = 6 + cos4x
[tex]\rm A(x) = \pi((6+cos4x)^2-6^2)[/tex]
[tex]\rm A(x) = \pi (cos^24x+12cos4x)[/tex]
[tex]\rm A(x) = \pi( \dfrac{1+cos8x}{2}+12cos4x)[/tex]
For the volume;
[tex]\rm Volume = \int\limits^\dfrac{\pi}{8}}_0 {A(x)} \, dx[/tex]
Put the function A(x) in the above integration and solve we will get:
[tex]\rm Volume = 3\pi + \dfrac{\pi^2}{16}[/tex]
Thus, the volume will be 3π + π²/16 if the solid obtained by rotating the region bounded by the given curves about the specified axis. Y=0, y= \cos(4 x), x = π/8, x = 0 about the axis y= -4f
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