Find the centroid of the region bounded by the curve sin x on the interval (0, pi) and the x-axis.
Please help
Let C(x,y)-centroid. Obviously, x=pi/2
The area of the region=2
y=(1/2)(1/2)(integral from 0 to pi) sin^2(x)dx=pi/8
To find the centroid of a region bounded by a curve and the x-axis, we need to find the x-coordinate and y-coordinate of the centroid.
Step 1: Determine the equation of the curve:
In this case, the curve is given by y = sin(x).
Step 2: Find the x-coordinate of the centroid:
The x-coordinate of the centroid is given by the formula:
x_bar = (1/A) * ∫[a, b] (x * f(x)) dx,
where A is the area of the region, and f(x) is the equation of the curve.
In our case, a = 0 and b = pi. Also, the area can be calculated as the integral of f(x) from a to b:
A = ∫[0, pi] f(x) dx.
So, the x-coordinate of the centroid is:
x_bar = (1/A) * ∫[0, pi] (x * sin(x)) dx.
Step 3: Find the y-coordinate of the centroid:
The y-coordinate of the centroid is given by the formula:
y_bar = (1/A) * ∫[a, b] (f(x)) dx.
In our case, the y-coordinate of the centroid is:
y_bar = (1/A) * ∫[0, pi] (sin(x)) dx.
Now, let's evaluate these integrals to find the x-coordinate and y-coordinate of the centroid.
Step 4: Evaluate the integrals:
To evaluate these integrals, we can use integration techniques or lookup tables.
∫[0, pi] (x * sin(x)) dx:
Using integration by parts, let u = x and dv = sin(x) dx:
du = dx and v = -cos(x).
Applying the integration by parts formula:
∫[0, pi] (x * sin(x)) dx = [-x * cos(x)] from 0 to pi + ∫[0, pi] (cos(x)) dx.
= [-pi * cos(pi) - 0 * cos(0)] + [sin(pi) - sin(0)]
= pi + 0 = pi.
∫[0, pi] (sin(x)) dx:
The integral of sin(x) is -cos(x).
∫[0, pi] (sin(x)) dx = [-cos(x)] from 0 to pi
= -cos(pi) - (-cos(0))
= 1 - (-1)
= 1 + 1 = 2.
Now, let's calculate the x-coordinate and y-coordinate of the centroid.
Step 5: Calculate the centroid:
x_bar = (1/A) * ∫[0, pi] (x * sin(x)) dx = (1/pi) * pi = 1.
y_bar = (1/A) * ∫[0, pi] (sin(x)) dx = (1/2) * 2 = 1.
Therefore, the centroid of the region bounded by the curve sin(x) on the interval (0, pi) and the x-axis is located at the point (1, 1).
To find the centroid of the region bounded by the curve sin(x) on the interval (0, π) and the x-axis, we can use the formula:
x̅ = (1/A) * ∫[a,b] x f(x) dx
y̅ = (1/A) * ∫[a,b] A f(x) dx
where:
x̅ is the x-coordinate of the centroid,
y̅ is the y-coordinate of the centroid,
A is the area of the region,
[a, b] is the interval over which we are finding the centroid, and
f(x) is the function that defines the curve.
In this case, f(x) = sin(x), and the interval is (0, π).
Let's start by finding the area of the region.
A = ∫[0,π] |f(x)| dx
= ∫[0,π] |sin(x)| dx
To calculate this integral, we can split it into two separate integrals, since the absolute value of sin(x) changes sign at π/2:
A = ∫[0,π/2] sin(x) dx + ∫[π/2,π] -sin(x) dx
Evaluating these integrals, we get:
A = [-cos(x)]|[0,π/2] + [-cos(x)]|[π/2,π]
= (-cos(π/2) - (-cos(0))) + (-cos(π) - (-cos(π/2)))
= (-0 - (-1)) + (1 - 0)
= 1 + 1
= 2
Now, let's find the x-coordinate of the centroid:
x̅ = (1/A) * ∫[0,π] x * sin(x) dx
To calculate this integral, we can integrate by parts, using u = x and dv = sin(x) dx:
x̅ = (1/A) * [x*(-cos(x)) - ∫[-cos(x)] dx]|[0,π] - (1/A) * ∫[0,π] -cos(x) dx
= (1/A) * (0 - 0) - (1/A) * [sin(x)]|[0,π]
= - (1/A) * (sin(π) - sin(0))
= - (1/A) * (0 - 0)
= 0
Therefore, the x-coordinate of the centroid is 0.
Finally, let's find the y-coordinate of the centroid:
y̅ = (1/A) * ∫[0,π] A * sin(x) dx
= (1/A) * A * [-cos(x)]|[0,π]
= 1
Therefore, the y-coordinate of the centroid is 1.
In conclusion, the centroid of the region bounded by the curve sin(x) on the interval (0, π) and the x-axis is (0, 1).