Find the specified coordinates of the centroid of the area bounded by
a) x=3 and 2x=y^2 find the coordinates x and y of the centroid
b) y= x^3 and y=4x in the first quadrant. Find the x coordinate of the centroid
To find the coordinates of the centroid of an area bounded by curves, we need to use the concept of integration. The centroid coordinates can be calculated using the formulas:
For x-coordinate of centroid (xc):
xc = (1/A) * ∫[a,b] (x * f(x))dx
where A is the area between the curves and [a, b] represents the interval where the curves cross each other.
For y-coordinate of centroid (yc):
yc = (1/A) * ∫[a,b] [(f(x))^2]dx
Let's calculate the coordinates:
a) For x=3 and 2x=y^2:
First, let's find the intersection points:
2x = y^2
2(3) = y^2
6 = y^2
y = ±√6
Since we are interested in the area, we'll consider the positive value of y, which is y = √6.
To find the x-coordinate of the centroid (xc), we need to determine the limits of integration:
The curve y = √6 intersects the line x = 3 at (3, √6) and (3, -√6).
xc = (1/A) * ∫[a,b] (x * f(x))dx
= (1/Area) * ∫[3,3] (x * √6)dx
= (1/Area) * 3 * √6
To find the y-coordinate of the centroid (yc), we use:
yc = (1/A) * ∫[a,b] [(f(x))^2]dx
= (1/Area) * ∫[3,3] (√6)^2dx
= (1/Area) * 6
We don't know the area, so we'll leave the answers in terms of A:
xc = (3 * √6) / A
yc = 6 / A
b) For y = x^3 and y = 4x in the first quadrant:
Let's find the intersection points:
x^3 = 4x
x^3 - 4x = 0
x(x^2 - 4) = 0
x(x - 2)(x + 2) = 0
The intersection points are at x = 0, x = 2, and x = -2. Since we are interested in the first quadrant, we consider x = 0 and x = 2.
To find the x-coordinate of the centroid (xc), we need to determine the limits of integration:
The curve y = x^3 intersects the line y = 4x at (0, 0) and (2, 8).
xc = (1/A) * ∫[a,b] (x * f(x))dx
= (1/Area) * ∫[0,2] (x * x^3)dx
= (1/Area) * ∫[0,2] x^4 dx
To find the y-coordinate of the centroid (yc), we use:
yc = (1/A) * ∫[a,b] [(f(x))^2]dx
= (1/Area) * ∫[0,2] (x^3)^2dx
= (1/Area) * ∫[0,2] x^6 dx
We don't know the area, so we'll leave the answers in terms of A:
xc = (∫[0,2] x^4 dx) / (∫[0,2] x^6 dx)
yc = (∫[0,2] x^6 dx) / (∫[0,2] x^6 dx)