# Area between 2 curves, Volume around X and Y, Centroid

• DanRow93
In summary, the conversation discussed finding the area between two curves, g(x) and f(x), and calculating the volume using integrals. The point of intersection was found and used to determine the area between the curves, which was then used to find the centroid of the area using the theorem of pappus. However, it is unclear why volumes were used instead of just working with the area and how π was involved in the calculations.
DanRow93
g(x)= √(19x) = upper curve
f(x)= 0.2x^2 = lower curve

Firstly, I found the point of intersection, which would later give the upper values for x and y.
x=7.802
y=12.174

Then I found the area under g(x) and took away the area under f(x) to get the area between the curves.
31.67 units^2

This is where I will include the equations so it is easy to see if I have done it right.

VolX = ∫π[(g(x)^2)-(f(x)^2)] dx
(The outer line squared minus the inner line squared)
Putting in my values for g(x)^2 and f(x)^2
g(x)^2 = 19x
f(x)^2 = 0.04x^4
Between 0 and 7.802 this gave me a volume of 1090.133 units^3

VolY = ∫π[(f(y)^2)-(g(y)^2)] dy
This time the line that was the inner in relation to the x-axis becomes the outer and vice versa.
f(y)^2 = 5y
g(y)^2 = (y^4)/361
Solving this integral between 0 and 12.174 gave me a volume of 698.59 units^3

Next, the task was to find the centroid of the area, specifically using the theorem of pappus. This task is given a distinction mark, however after looking at it it seems quite simple and I'm not sure if I am missing something.

VolX = Shaded area x 2π(ybar)

So (1090.133)/(31.67x2π) = ybar = 5.478

Then I haven't done xbar yet but I would think that I just do the same with VolY = Shaded area x 2π(xbar)

Is there anything that you can see that I might have missed that makes the last part more difficult?

Thanks!

DanRow93 said:
the area under g(x)
I gather it is also bounded by x=0.
DanRow93 said:
centroid of the area
DanRow93 said:
VolX =
I don't understand why you are working with volumes to find the centroid of an area. In particular, how do you get π in there?

## 1. What is the formula for finding the area between two curves?

The formula for finding the area between two curves is ∫(upper curve - lower curve) dx. This means taking the integral of the difference between the upper and lower curves with respect to x.

## 2. How do you find the volume around X and Y?

To find the volume around X and Y, you can use the formula V = ∫A(x) dx or V = ∫A(y) dy, where A(x) and A(y) are the cross-sectional areas perpendicular to the x and y axes, respectively. These integrals will give you the volume of the solid formed by rotating the area between two curves around the x or y axis.

## 3. What is the centroid of a shape?

The centroid of a shape is the point where all the mass of the object is evenly distributed. In terms of the area between two curves, the centroid is the point where the two curves have equal areas on either side.

## 4. How do you find the centroid of a region?

To find the centroid of a region, you can use the formula x̄ = (1/A)∫x dA or ȳ = (1/A)∫y dA, where A is the total area of the region and x and y are the coordinates of each point in the region. These integrals will give you the x and y coordinates of the centroid.

## 5. Can you use the centroid formula for irregular shapes?

Yes, the centroid formula can be used for irregular shapes. However, it may be more difficult to find the integrals needed to calculate the coordinates of the centroid. In these cases, it may be helpful to break the shape into smaller, more regular shapes and use the centroid formula for each individual shape.

• Calculus and Beyond Homework Help
Replies
21
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
4K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus
Replies
20
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
4K
• Calculus
Replies
2
Views
2K
• Calculus and Beyond Homework Help
Replies
4
Views
1K