Find the centroid (¯ x, ¯ y) of the region bounded by:
y = 6x^2+7x, y = 0, x = 0,
and x = 7
shut up steve they asked for the answer not for a link to a similar problem
Why did the centroid go to therapy? Because it couldn't find its center!
To find the centroid of the region bounded by the given equations, we need to calculate the average x-coordinate (¯ x) and the average y-coordinate (¯ y) of the region.
First, let's find the x-coordinate of the centroid (¯ x):
∫[0 to 7] x*(6x^2 + 7x) dx
Calculating this integral will give us the value of ¯ x.
Next, let's find the y-coordinate of the centroid (¯ y).
∫[0 to 7] (1/2)(6x^2 + 7x)^2 dx / ∫[0 to 7] (6x^2 + 7x) dx
After evaluating the integrals, we'll get the value of ¯ y.
So don't worry, the centroid will eventually find its balance!
To find the centroid (x̄, ȳ) of the region bounded by the given curves, you need to calculate the following:
1. Find the area (A) of the region bounded by the curves.
2. Calculate the x-coordinate of the centroid (x̄) using the formula:
x̄ = (1 / A) * ∫[a,b] (x * f(x)) dx
where [a, b] represents the interval of integration and f(x) is the equation of the curve.
3. Calculate the y-coordinate of the centroid (ȳ) using the formula:
ȳ = (1 / 2A) * ∫[a,b] (f(x))^2 dx
Let's apply these steps to find the centroid (x̄, ȳ):
Step 1: Finding the Area (A)
To find the area (A) bounded by the curves, you need to calculate the definite integral of the difference between the two curves:
A = ∫[a,b] (f(x) - g(x)) dx
In this case, the two curves are y = 6x^2 + 7x and y = 0, and the interval of integration is [0, 7]. Therefore:
A = ∫[0,7] (6x^2 + 7x - 0) dx
Step 2: Calculating x-coordinate of the Centroid (x̄)
To find the x-coordinate of the centroid (x̄), use the formula:
x̄ = (1 / A) * ∫[a,b] (x * f(x)) dx
In this case, f(x) = 6x^2 + 7x. Therefore:
x̄ = (1 / A) * ∫[0,7] (x * (6x^2 + 7x)) dx
Step 3: Calculating y-coordinate of the Centroid (ȳ)
To find the y-coordinate of the centroid (ȳ), use the formula:
ȳ = (1 / 2A) * ∫[a,b] (f(x))^2 dx
In this case, f(x) = 6x^2 + 7x. Therefore:
ȳ = (1 / 2A) * ∫[0,7] ((6x^2 + 7x)^2) dx
Performing the necessary calculations will give you the values of x̄ and ȳ, which represent the x-coordinate and y-coordinate of the centroid (¯ x, ¯ y) of the region bounded by the given curves.
See here:
http://www.intmath.com/applications-integration/5-centroid-area.php
scroll down for a nice example, with steps for solution.