Centroid:
Find the centroid of the region cut from the f...

Question

Centroid:

Find the centroid of the region cut from the first quadrant by the curve

y = 1/√(x + 1) and the line x = 3.

Accepted Answer

Welcome back everyone. In this problem we want to find the x coordinate of the centroid of the region r located in the first quadrant that is bounded above by the curve y equals 1 divided by root x + 2 on the right by the vertical line x equals 2 and below by the x axis. A says it's 2/3 of route 2, B 4/3 of route 2, C 1/2 of route 3, and D 3/5 of route 3. Now let's quickly try to sketch what's going on here to get an idea of what we're looking for. OK, so if I put our x and y axis here, OK, then we're looking for the centroid of the region r in the first quadrant that's bounded by the curvey, which would look something like this, OK, on the right by the vertical line x equals 2, which here I could represent that as this dotted line, OK. So this is the line x equals 2 and below by the x axis, OK, so we already have our x axis here that I can label as X. So basically this region R is right here, which I can represent with uh shading it in red, OK. So this is the region R in which we're looking for the x coordinate of its centroid, OK. So if I, if I were to Take out. Let me just put R somewhere else here. OK, and let's say that this is the centroid, then we're looking for the X coordinate of the centroid. So how can we figure it out? What do we know? Well, notice that if we were to define the region R. It's equal to, OK, for its x y coordinates x being between 0 and 2, OK, including 0 and 2, and its y value or its y values are between 0 and or curve y. So that's 1 divided by root x plus 2, OK. And now in this case, recall that if we were to figure out over the x coordinate of our centroid. It will be equal to the first moment about the y axis i x divided by the area of our region r which is a. So if we can figure out what both of these are, then we should be able to solve for the x coordinate of the centroid. So let's first, let's start by calculating the area. What do we know? Well, for the area, recall that it's equal to the definite integral. Between the limits of our interval of y with respect to x, so in this case that means the area will be equal to the definite integral between 0 and 2 because remember x is bounded by 0 and 2 of or curve 1 divided by the square root of x + 2 with respect to x. So let's go ahead and integrate this expression. Let's integrate using substitution. So let's let u be equal to x + 2, which means that d u will be equal to dx. So that means if we take our indefinite integral, OK, that's 1, the integral of 1 divided by the spirit of x plus 2 with respect to x, then rewriting it in terms of u, it will be the integral of 1 divided by root u with respect to u. Which we could reread as the integral of u to the negative half with respect to u, which if we integrate that it's going to be equal to 2 multiplied by a root u plus c. And since we know what u is, we can replace it here to get this to be 2 multiplied by a root x plus 2 plus c. Thus this tells us then that the area will be equal to 2 multiplied by root x + 2 evaluated between the bones of 0 and 2 which would be equal to 2 root 2 + 2 minus 2 root 0 + 2. Now root 2 + 2 is root 4, and that's 2 and 2 multiplied by 2 is 4, whereas 0 + 2 is 2, so we're left with -2 root 2. So this is our area 4 minus 2 root 2 square units. That's the area of the region r. Now let's see if we can figure out the first moment about the y axis. How can we get that? Well, recall, OK, that. The moment about the y axis will be equal to the definite integral between the bonds of our interval a and b of x multiplied by our function f x with respect to x. So in this case that's going to be equal to the definite integral between 0 and 2 of x multiplied by 1 divided by the square root of x + 2 with respect to x. Now again, let's see if we can integrate using substitution. So let's let you be equal to x + 2. Then that means x would be equal to u minus 2 and the derivative of u with respect to x would be 1, so d u would be equal to dx if we were to change the limits for you when x equals 0, u would be equal to 2. And when X equals 2. Then you equals 4. So no, this means then that our moment. Will be equal to the integral between 2 and 4 of u minus 2 multiplied by u to the negative12. With respect to you Now let's try to simplify here. OK, if we distribute you to the negative half across our terms, then this will become u12 minus 2u to the negative1 with respect to you. Next, integrating both of those terms, it will become 2/3 of u to the power of 3 halves minus 4 u 1/2, OK, between the bones of 2 and 4. And now when we apply our upper and substitute our upper and lower bones, then we'll get. OK, we will end up with 2/3, OK, of 43 halves minus 4 multiplied by 4/22. Minus 2/3 multiplied by 23 halves minus 4 multiplied by 2 1/2. Now, remember, in our result and for our area we had a root 2 in there, so we want to try to simplify our expression to somehow include that. Notice that our powers here are fractional, but at some point we want to simplify to include the root. So if we take it a step further, OK, then this is going to become 2/3. Multiplied by 8 minus 8 because 452 is 2 and 4 multiplied by 2 is 8, OK, minus 2/3 multiplied by 22 minus 42. I know if we rewrite this entire expression with a common denominator of 3, it will become 16 minus 24 minus 42 minus 12 2. All divided by 3. Which will simplify to -8 minus 82 or plus 8 2 divided by 3, which then we can rewrite as 8 multiplied by root 2 minus 1 if we factor out 8 divided by 3. So this is the first moment about the y axis. Know that we have our first moment and the area, then we can find the x coordinate of the centroid because no, that means it will be equal to our moment 8 root 2 minus 1. OK, or 8 multiplied by root 2 minus 1. OK, divided by 3 divided by our area of 4 minus 2 root 2, which we can rewrite as 3 multiplied by 4 minus 2 root 2 in our denominator, and when we write it like that, it will be easier to see how we can rationalize it. So if we multiply here by 4 root 2 divided by 4 plus root. Let me, my apologies here, 4 + 2 root 2. I'm running ahead of myself, divided by 4 + 2 root 2, which if we simplify. Then in our numerator, we'll have 8 multiplied by 4 root 2 plus 4 minus 4 minus 2 root 2. OK. All divided by 3 multiplied by 16 minus 8. And when we simplify here, we get the X coordinate to be 2 root 2 divided by 3. So the X coordinate of the centroid of the region R is 2/3 of root 2. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Centroid:Find the centroid of the region cut from the first quadrant by the curvey = 1/√(x + 1) and the line x = 3.

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Centroid:
Find the centroid of the region cut from the first quadrant by the curve
y = 1/√(x + 1) and the line x = 3.