Finding the Centroid of a Shaded Area with y=sqrt(x) Curve: Step-by-Step Guide"

[Bradracer18]

Homework Statement

I need to locate the centroid of the shaded area, in my picture. The shaded area is under the line(in between the x-axis and the curve.)

The curve is y=sqrt(x) And stretches a length from the y axis(0,0), along the x axis, a length of b.

This is a new concept, so I'm not really familiar with how to do these. It will probably take some walking through, for me to understand it.

Homework Equations

A = Int(dA)...not sure how to use integral, so will just use int.

X = int(xdA)

The Attempt at a Solution

I'm not sure how to find dA. I think I need to take a slice(or maybe 2) of the curved area. I'm not sure how to come up with those equations.

View attachment problem 5.bmp

 

[Nate810]

Then, I think that I need to find xdA, which I'm also not sure how to come up with. I'm assuming it's something like y times dx. But again, I'm not sure how to come up with the equation for that. After that, I think I would just integrate both of those equations, from 0 to b, and then divide the X equation by the A equation. Is this the correct approach? If so, what equation should I use for dA and xdA? Any help would be greatly appreciated. Thank you.

 

[m2003]

I would first like to clarify that finding the centroid of a shaded area is a mathematical concept, not a scientific one. However, as a scientist, I can offer some guidance on how to approach this problem.

Firstly, the concept of a centroid refers to the center of mass of a shape or object. In this case, we are looking for the center of mass of the shaded area under the curve y=sqrt(x). This is important to keep in mind as we go through the steps.

Step 1: Understanding the problem

The first step is to understand the problem and what is being asked. From the given information, we know that we need to find the centroid of the shaded area under the curve y=sqrt(x). This area is bounded by the x-axis and the curve, and has a length of b along the x-axis.

Step 2: Sketch the problem

It is always helpful to visualize the problem by sketching it out. Draw a coordinate system with the x-axis and y-axis, and label the points (0,0) and (b, 0) to represent the boundaries of the shaded area.

Step 3: Determine the equation for the shaded area

We can use the equation for the curve y=sqrt(x) to determine the equation for the shaded area. The shaded area is bounded by the x-axis, so the lower limit of integration will be 0. The upper limit of integration will be b, as this is the length of the shaded area along the x-axis. Therefore, the equation for the shaded area can be written as:

A = ∫0b sqrt(x) dx

Step 4: Evaluate the integral

Integrating the equation from step 3 will give us the area of the shaded region. This is an important step as it will help us determine the weight of each infinitesimal slice of the shaded area, which is necessary for finding the centroid.

A = ∫0b sqrt(x) dx = (2/3)*b^(3/2)

Step 5: Find the weight of each infinitesimal slice

To find the weight of each infinitesimal slice, we need to divide the total area by the length of the shaded area. This will give us the weight per unit length.

w = (2/3)*b^(3/2) / b = (2/3)*b^(1/2)

Step 6: Determine the x