Centroid of a region
Find the centroid of the region in ...

Question

Centroid of a region

Find the centroid of the region in the plane enclosed by the curves y = ±(1 − x²)^(-1/2) and the lines x = 0 and x = 1.

Accepted Answer

Find the centroid X Y bar of the region and the XY plane bounded by the curves Y equals plus or minus 1 divided by 1 + X2 and the vertical lines X equals 0 and X equals 1. We have 4 possible answers, which are points of the synthroid, 2, natural log 2, divided by pi, 04, natural log 2 divided by pi, 0, i, natural log 2, divided by 4, pi divided by 4, or pi natural log 2 divided by 8, pi divided by 4. Now, we notice here, the top and bottom boundaries are exact reflections of each other because of the plus and minus. That means the Y will be right in the middle, Y bar being 0. Now we just need to find x bar for the centroid. To do this, we will use X equals MY divided by A. Where MY will be the integral. From A to B Of X multiplied. By Y top minus Y bottom. D X Now A will just be the integral. Of Y top minus y bottom. This is with DX. Now, it tells us we are from A to B, which in our case, A is 0, B is 1. We'll find a first. We have the integral from 0 to 1. Of 1 divided by 1 + X2 minus -1 divided by 1 + X2. D x, which is just the integral from 0 to 1. Of 2 divided by 1 plus X2. Now, this is actually a known integral, which is the arc tangent or tangent inverse. This gives us to tangent inverse of X from 0 to 1. We can plug in our values to simplify this and give us pi divided by 2. Now let's find our moments in white. MY will be given by the integral from 0 to 1 of x multiplied by a difference, which we noted as 2 divided by 1 plus x 2. This is with TX at the end. Now we can use a use substitution for this. We'll first simplify this to be the integral from 0 to 1 of 2 x divided by 1 plus x2 dx. Well, that u equal a denominator 1 + x2, and duu is the derivative of 1 + X2, 2 x. D X We notice then that duu replaces the two x in the numerator and the dx. We get the integral. Of dx divided by You, and it's actually not DX, that is now DU. Now our bounds will be different. Because U equals 1 + X2. We have u equals 1 + 02 and u equals 1 + 12, giving us 1 to 2. We now have the integral from 1 to 2 of 1 divided by u to u. Which is just natural log of you. From 1 to 2. This just simplifies to natural log of 2, because natural log of 1 goes to 0. We can now calculate our moment. And centroid value for X. X bar will then be MY natural log 2. Divided by pi divided by 2. Which is simplifies to 2 natural like 2 divided by pi. That means our synthroid, X Y bar will be 2, log 2, divided by pi, 0. This is the answer to our problem. If we look at our answers. This is answer A. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Centroid of a regionFind the centroid of the region in the plane enclosed by the curves y = ±(1 − x²)^(-1/2) and the lines x = 0 and x = 1.

Watch next

Centroid of a region
Find the centroid of the region in the plane enclosed by the curves y = ±(1 − x²)^(-1/2) and the lines x = 0 and x = 1.