Answer:
5.81
Step-by-step explanation:
The method of cylindrical shells uses this formula:
[tex]\int\limits^a_b {2 \pi * shellRadius * shellHeight} \, dx[/tex]
It integrates the pieces of area in the region.
In this case:
[tex]2 \pi * \int\limits^2_0 {x * cos(\frac{\pi}{4} * x)} \, dx[/tex]
Integrating by computer:
[tex]2*\pi*(\frac{4}{\pi^2}*(\pi*x*sin(\frac{\pi}{4}*x) + 4*cos(\frac{\pi}{4}*x)))[/tex]
Evaluating between 0 and 2
[tex]2\pi * (\frac{4}{\pi^2}*(\pi*2*sin(\frac{\pi}{4}*2) + 4*cos(\frac{\pi}{4}*2)) - \frac{4}{\pi^2}*(\pi*0*sin(\frac{\pi}{4}*0) + 4*cos(\frac{\pi}{4}*0))) = 5.81[/tex]
The volume V generated by rotating the region bounded by the curves about the given axis is 5.714 unit³.
According to the method of cylindrical shells,
[tex]V = \int^a_b 2\pi \times \rm shell\ Radius \times shell\ height\ dx[/tex]
Given to us,
y = cos(πx/4), y = 0, 0 ≤ x ≤ 2; about the y-axis
We know the formula of cylindrical shells,
[tex]V = \int^a_b 2\pi \times \rm shell\ Radius \times shell\ height\ dx[/tex]
Substitute the values,
[tex]V = \int^2_0 2\pi \times x\times cos(\dfrac{\pi x}{4})\ dx[/tex]
[tex]=8x\ sin\ (\dfrac{\pi x}{4}) + \dfrac{32\ cos(\dfrac{\pi x}{4})}{\pi}+c\\\\\\= 2(\dfrac{8}{\pi}-\dfrac{16}{\pi^2})\pi\\\\\\= 5.814[/tex]
Hence, the volume V generated by rotating the region bounded by the curves about the given axis is 5.714 unit³.
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