Find the centroid of the area bounded by the curves:
sqrt(x)+sqrt(y)=sqrt(a)
x=0
y=0
To find the centroid of the area bounded by the curves, we need to use the concept of the double integral. Let's break down the steps to find the centroid.
Step 1: Determine the limits of integration
From the given information, we have the following curves bounding the area:
1. The line x = 0
2. The line y = 0
3. The curve sqrt(x) + sqrt(y) = sqrt(a)
To determine the limits of integration, we need to find the points of intersection between the curves. Let's solve for the intersection points:
For the first intersection point:
sqrt(x) + sqrt(y) = sqrt(a)
When x = 0, the equation becomes:
sqrt(y) = sqrt(a)
Squaring both sides, we get:
y = a
So, the first intersection point is (0, a).
For the second intersection point:
sqrt(x) + sqrt(y) = sqrt(a)
When y = 0, the equation becomes:
sqrt(x) = sqrt(a)
Squaring both sides, we get:
x = a
So, the second intersection point is (a, 0).
Now, we have determined the limits of integration for x and y:
x ranges from 0 to a, and y ranges from 0 to a.
Step 2: Set up the double integral
The centroid of the area is given by the following formulas:
x-bar = (1/A) * ∬(x * f(x, y)) dA
y-bar = (1/A) * ∬(y * f(x, y)) dA
Where A is the area of the bounded region and f(x, y) represents the given curve.
In this case, f(x, y) = sqrt(x) + sqrt(y) - sqrt(a).
Step 3: Calculate the area A
To calculate the area A, we can integrate the area element dA over the given region.
A = ∬ dA, where the limits of integration are x: 0 to a and y: 0 to a.
The double integral of dA simplifies to the product of the range of x and y:
A = ∫[0,a] ∫[0,a] dx dy
Step 4: Calculate the centroid coordinates
Now, we can evaluate the centroid coordinates using the formulas mentioned earlier:
x-bar = (1/A) * ∬(x * f(x, y)) dA
y-bar = (1/A) * ∬(y * f(x, y)) dA
Plug in f(x, y) = sqrt(x) + sqrt(y) - sqrt(a) and evaluate the double integrals.
After evaluating the double integrals, we will have the x-bar and y-bar coordinates of the centroid of the bounded region.