Find the coordinates of the centroid of the regions bounded by the graphs of:
i)y=x2+1,x=0,x=1 and y=0
ii)y+x2=6 and y+2x-3=0
I'll show you how to do one of them.
i) For the x coordinate of the centroid, integrate
(Integral of) x(x^2+1) dx from x=0 to x=1
= [x^4/4 +x^2/2]@x=1 - [x^4/4 -x^2/2]@x=0
= 1/4 + 1/2 = 3/4
For the y coordinate of the centroid, integrate
(Integral of) (y/2)(x^2+1) dx from x=0 to x=1
= (Integral of) (1/2)(x^2+1)^2 dx from x=0 to x=1
= (Integral of) (1/2)(x^4+2x^2+1)^2 dx from x=0 to x=1
= (1/2)[x^5/5 +(2/3)x^3 +x] @ x=1
- (1/2)[x^5/5 +(2/3)x^3 +x] @ x=0
= 1/10 + 1/3 + 1/2 = 14/15
To find the coordinates of the centroid of the regions bounded by the given graphs, we need to calculate the x-coordinate and y-coordinate of the centroid separately and then combine them to get the centroid coordinates.
i) For the graph y = x^2 + 1, x = 0, x = 1, and y = 0:
1. Calculate the x-coordinate of the centroid:
The formula for the x-coordinate of the centroid is given by:
x-coordinate = (1/A) * ∫[a,b] x * f(x) dx
where A is the area of the region bounded by the graphs, and f(x) represents the equation of the curve.
To calculate the area A, we integrate the function f(x) - g(x) with respect to x, where f(x) is the upper curve (y = x^2 + 1) and g(x) is the lower curve (y = 0).
A = ∫[0,1] (x^2 + 1 - 0) dx
= ∫[0,1] (x^2 + 1) dx
= [x^3/3 + x] from 0 to 1
= (1/3 + 1) - (0/3 + 0)
= 4/3
Now, calculate the integral for the x-coordinate:
x-coordinate = (1/A) * ∫[a,b] x * (f(x) - g(x)) dx
= (3/4) * ∫[0,1] x * (x^2 + 1) dx
= (3/4) * ∫[0,1] (x^3 + x) dx
= (3/4) * [(x^4/4) + (x^2/2)] from 0 to 1
= (3/4) * [(1/4) + (1/2)] - [(0/4) + (0/2)]
= (3/4) * (3/4)
= 9/16
So, the x-coordinate of the centroid is 9/16.
2. Calculate the y-coordinate of the centroid:
The formula for the y-coordinate of the centroid is given by:
y-coordinate = (1/2A) * ∫[a,b] f(x)^2 dx
Now, calculate the integral for the y-coordinate:
y-coordinate = (1/(2*A)) * ∫[0,1] (x^2 + 1)^2 dx
= (1/(2*(4/3))) * ∫[0,1] (x^4 + 2x^2 + 1) dx
= (3/8) * [(x^5/5) + (2x^3/3) + x] from 0 to 1
= (3/8) * [(1/5) + (2/3) + 1] - [(0/5) + (0/3) + 0]
= (3/8) * (28/15)
= 7/10
So, the y-coordinate of the centroid is 7/10.
Therefore, the coordinates of the centroid of the region bounded by the graphs in this case are (9/16, 7/10).
ii) For the graphs y + x^2 = 6 and y + 2x - 3 = 0:
To find the centroid of this region, we need to solve the two equations simultaneously to find the intersection points. Then, we calculate the average of the x-coordinates and y-coordinates of these intersection points to get the centroid coordinates.
First, solve the two equations simultaneously:
y + x^2 = 6 ...(1)
y + 2x - 3 = 0 ...(2)
From equation (2), we can express y in terms of x as:
y = -2x + 3
Substitute this value of y into equation (1):
-2x + 3 + x^2 = 6
Rearrange the equation:
x^2 - 2x - 3 = 0
Factorize the equation:
(x - 3)(x + 1) = 0
Solving the equation gives two solutions:
x = 3 and x = -1
Now, substitute the values of x back into equation (2) to find the corresponding y-coordinates:
For x = 3:
y + 2(3) - 3 = 0
y + 3 = 0
y = -3
For x = -1:
y + 2(-1) - 3 = 0
y - 2 - 3 = 0
y - 5 = 0
y = 5
So, the intersection points are (3, -3) and (-1, 5).
To find the centroid coordinates, we calculate the average of the x-coordinates and y-coordinates of the intersection points:
x-coordinate of centroid = (3 + (-1))/2 = 1/2
y-coordinate of centroid = (-3 + 5)/2 = 1
Therefore, the centroid coordinates of the region bounded by the graphs in this case are (1/2, 1).
To find the coordinates of the centroid of a region bounded by the graphs of certain equations, we need to calculate the coordinates of the centroid using the equations for finding the x-coordinate and y-coordinate of the centroid. The formulas are as follows:
x-coordinate of centroid:
x̄ = (1/A) * ∫[a,b] (x * f(x)) dx
y-coordinate of centroid:
ȳ = (1/A) * ∫[a,b] (0.5 * (f(x))^2) dx
where A is the area of the region, f(x) represents the given equation, and [a, b] is the interval on which the region lies.
Let's apply these formulas to the given regions:
i) For the region bounded by y = x^2 + 1, x = 0, x = 1, and y = 0:
First, find the points of intersection to determine the interval [a, b]:
To find the points of intersection of y = x^2 + 1 and y = 0, solve x^2 + 1 = 0.
This quadratic equation has no real solutions since x^2 cannot be negative, so there is no intersection point where y = 0.
Next, calculate the x-coordinate of the centroid:
Since there is no region between the curves and the x-axis, the x-coordinate of the centroid is undefined.
Similarly, the y-coordinate of the centroid is also undefined.
Therefore, for this region, the centroid cannot be determined.
ii) For the region bounded by y + x^2 = 6 and y + 2x - 3 = 0:
First, express y in terms of x:
From the second equation, y = 3 - 2x.
Next, find the points of intersection to determine the interval [a, b]:
Solve the system of equations:
y + x^2 = 6
y + 2x - 3 = 0
Substituting the value of y from the second equation into the first equation, we get:
3 - 2x + x^2 = 6
Rearranging the equation, we have:
x^2 - 2x - 3 = 0
Factoring the quadratic equation, we get:
(x - 3)(x + 1) = 0
So, x = 3 or x = -1. These are the points of intersection.
Now, calculate the x-coordinate of the centroid:
To find the x-coordinate of the centroid, calculate the definite integral of (x * f(x)) from x = -1 to x = 3 and divide it by the area of the region.
A = ∫[-1,3] (x + 2x^2 - 3x) dx
= ∫[-1,3] (-x^2 - 2x) dx
= -∫[-1,3] (x^2 + 2x) dx
Next, calculate the y-coordinate of the centroid:
To find the y-coordinate of the centroid, calculate the definite integral of (0.5 * (f(x))^2) dx from x = -1 to x = 3 and divide it by the area of the region.
ȳ = (1/A) * ∫[-1,3] (0.5 * ((3 - 2x)^2)) dx
Evaluate the integrals and divide them by the area, A, to get the x-coordinate and y-coordinate of the centroid.
Therefore, to find the coordinates of the centroid of the region bounded by the graphs of y + x^2 = 6 and y + 2x - 3 = 0, you need to evaluate the definite integrals and divide them by the area of the region.