Find the centriod of the boundEd by the curvez y=x^2 and y=2x+3
The area of the region is
A = ∫[-1,3] (2x+3)-x^2 dx
= -1/3 x^3 + x^2 + 3x [-1,3]
= (-9+9+9)-(1/3 + 1 - 3)
= 32/3
For xbar, find
X = ∫[-1,3] x((2x+3)-x^2) dx
= -1/4 x^4 + 2/3 x^3 + 3/2 x^2 [-1,3]
= (-81/4 + 18 + 27/2)-(-1/4 + 2/3 + 3/2)
= 32/3
For ybar, find
Y = ∫[-1,3] 1/2 ((2x+3)^2 - (x^2)^2) dx
= -1/5 x^5 + 4/3 x^3 + 6x^2 + 9x [-1,3]
= 544/15
xbar = X/A = 1
ybar = Y/A = 17/5
To solve this category of problems where you need the area/centroid of a region bounded by two curves, you need to first find TWO intersection points of the two curves by equating y1(x)=x^2 and y2(x)=2*x+3.
The intersection points are at x=-1 and x=3, with y2(x) above y1(x).
Then you need to find the area by integrating
A=∫(y2(x)-y1(x))dx between the limits x=-1 and x=3.
=32/3
To find the centroid(xBar,yBar), you need to find the first moments,
xBar=∫x(y2(x)-y1(x))dx / A
and
yBar=∫(1/2)(y2(x)+y1(x))(y2(x)-y1(x))dx /A
from which I get
(xBar,yBar)=(1,17/5)
To find the centroid of the region bounded by the curves y = x^2 and y = 2x + 3, we need to calculate the coordinates (x̄, ȳ) of the centroid. The x-coordinate of the centroid (x̄) can be found by calculating the average of the x-coordinates of the points where the two curves intersect. The y-coordinate of the centroid (ȳ) can be obtained by calculating the average of the y-values of the region bounded by the curves.
First, let's find the points of intersection between the two curves:
By setting y = x^2 equal to y = 2x + 3, we can solve for x:
x^2 = 2x + 3
x^2 - 2x - 3 = 0
This is a quadratic equation. We can solve it by factoring or using the quadratic formula. Factoring the equation, we get:
(x + 1)(x - 3) = 0
Setting each factor equal to zero and solving for x, we find two solutions:
x + 1 = 0 --> x = -1
x - 3 = 0 --> x = 3
So, the two curves intersect at x = -1 and x = 3.
Next, we need to find the y-values of the points of intersection. We can substitute the x-values into either of the original equations to find the corresponding y-values.
For y = x^2:
When x = -1, y = (-1)^2 = 1
When x = 3, y = (3)^2 = 9
Now, we can calculate the x-coordinate of the centroid (x̄) using the formula:
x̄ = (1/area) * ∫[a, b] (x * f(x)) dx
In this case, the area is given by:
A = ∫[a, b] (f(x) - g(x)) dx
Where f(x) and g(x) are the two curves, and a and b are the x-values of the points of intersection.
To calculate the y-coordinate of the centroid (ȳ), we will use the formula:
ȳ = (1/area) * ∫[a, b] (F(x) - G(x)) dx
Where F(x) and G(x) are the integrals of the curves y = x^2 and y = 2x + 3, respectively.
Integrating both curves with respect to x, we have:
∫(x^2) dx = (1/3)x^3 + C1
∫(2x + 3) dx = x^2 + 3x + C2
Now, we have all the components to calculate the centroid coordinates. Plug in the values into the formulas mentioned earlier, and you will get the exact coordinates of the centroid for the region bounded by the given curves.