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A cosine is just a sine shifted to the left by π/2. A cosine of 4x is shifted to the left by only π/8 because of the factor 4. Sketch them.

The region we're looking for is this sausage-shaped part between the cos and the sin.

The x intercepts are at π/8 for the cosine and π/4 for the sine. The midpoint between them is at (π/8 + π/4)/2 = 3/16π.

The region is point symmetric around the x axis, so the y coordinate of the centroid is 0.

So the centroid is at (3/16π, 0)
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The centroid of the region bounded by the given curves y=8sin(4x), y=8cos(4x) is:  (3/16π, 0). See the definition of Centroid below.

What is a Centroid?

The center of mass of a shape or object that is geometric in nature and that has uniform density is called a centroid.

How do we obtain the centroid of the regions defined above?

Recall that when we shift a cosine to the left by π/2 it becomes a sine. Hence, a cosine of 4x is one that is moved to the left by only π/8

From the above, we thus know that the x-intercepts are at

  • π/4 for the sine,
  • π/8 for the cosine; and
  • the midpoint between them is at (π/8 + π/4)/2.

hence  (π/8 + π/4)/2 = 3/16π.

Because the region is point symmetric about the x-axis, the centroid's y-coordinate is 0.

So the centroid is at (3/16π, 0)

Learn more about Centroid at

https://brainly.com/question/7725518

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