20. Solid of revolution The region between the curve y=1/(2√x)...

Question

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.

b. Find the centroid of the region.

Accepted Answer

Find the centroid of the planar region bounded by the curve, Y equals 1 divided by the square root of X and the x axis for x between 1 and 9. Now, the centroid is given by the point X Y bar, where X bar is given by MY divided by A and Y bar is given by MX divided by A. Now A is the area. Which resist the integral from A to B. Of F of X D X. In this case, f x is 1 divided by the squared of x or x 1/2. So, we have the integral from 1 to 9 of X 1/2 DX. Now we can just solve this integral. This is the power rule, which means we have x 1/2 divided by 1/2 from 1 to 9. This is 2 multiplied by 9 1/2 minus 2 multiplied by 1 1/2. Which gives us A value of 4. Now we have A, we can find MY and MX. In this case, MY. Is given by The integral from 1 to 9 of X multiplied by F X DX. We have the integral from 1 to 9 of X multiplied by x to the -1/2 DX. Which is just the integral from 1 to 9 of X to the positive 1/2 DX by exponent rules. This gives us X to 3 halves divided by 3 halves from 1 to 9. If we plug in our bounce for this, we get 52 divided by 3. Now we can find MX. M xin is the integral from 1 to 9. Of 1/2 Multiplied By F of X 2 dx, this will be 1/2 integral from 1 to 9. Of x to the 1/2. Squared d x This will just give us 1/2 integral from 1 to 9 of x to the negative1 power, which is just 1 divided by XDX. Now we know this integral to be the natural log, so we get 1/2 natural log of X from 1 to 9. We can simplify this to get a value of natural log of 3. We can now use both of these, or our synthroid. You know, X bar Is MY divided by A, which is just 52/3 divided by 4. Y bar will be a natural log of 3 divided by 4. We can now simplify this. X bar, Y bar, the synthroid, isus, 13 divided by 3, and natural log of 3 divided by 4. This is the answer to our problem. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.b. Find the centroid of the region.

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20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
b. Find the centroid of the region.