find the coordinates of the centroid bounded by y=25x^2 and y=4. the region is covered by a thin flat plate
Last time I did this type of question was 55 years ago, so I hope I can remember.
I think we first have to find the area
intercepts between y = 25x^2 and y = 4
25x^2 = 4
x = ± 2/5
by symmetry we can go from 0 to 2/5 and double
area = 2(integral) (4 - 25x^2) dx from 0 to 2/5
= 2[ 4x - (25/3)x^3] from 0 to 2/5
= 2[ 8/5 - (25/3)(8/125) - 0 ]
= 2[ 16/15]
= 32/15
check my arithmetic
the x-coordinate
= (1/(32/15)) integral x(f(x)) dx from -2/5 to 2/5
= (15/32) integral 25x^3 dx from -2/5 to 2/5
= (15/32)[(25/4)x^4] from -2/5 to 2/5
= (15/32) ((25/4)(16/625) - (25/4)(16/625))
= 0 , ahhh, of course, look at the symmetry
for the y-coordinate:
= (1/(32/15)) (integral) (4^2 - (f(x)^2) /2 dx from -2/5 to 2/5
= (15/32) [8x - 125x^5/2] from -2/5 to 2/5
= (15/32) (16/5 - 125(32/3125)/2 - (-16/5 - 125(-32/3125)/2)
= (15/32)(32/5 + 8000/6250)
= 18/5 or 3.6
the centroid is (0,3.6)
according to my sketch this looks reasonable, most of the mass is towards the line y = 4
that does not look right according to my sketch
forget the very last line, I had made corrections, but forgot to delete that line
To find the coordinates of the centroid of the region bounded by the curves y = 25x^2 and y = 4, we can use the concept of integration.
Step 1: Find the intersection points of the curves:
Set the two equations equal to each other:
25x^2 = 4
Solve for x:
x^2 = 4/25
x = ± 2/5
Step 2: Determine the limits of integration:
To find the x-coordinates of the centroid, we need to integrate with respect to x. The limits of integration will be the x-values where the two curves intersect, which are x = -2/5 and x = 2/5.
Step 3: Set up the integral for the x-coordinate of the centroid:
The x-coordinate of the centroid (Cx) is given by the formula:
Cx = (1/A) ∫[b,a] x*f(x) dx
Here, f(x) represents the top curve minus the bottom curve, so it is (25x^2 - 4), and A represents the area of the region.
Step 4: Calculate the area of the region:
The area (A) of the region can be obtained by integrating the difference between the two curves over the limits of integration:
A = ∫[b,a] (25x^2 - 4) dx
Step 5: Evaluate the integral to calculate the area:
∫[b,a] (25x^2 - 4) dx = [ (25/3)x^3 - 4x ] [b,a]
Evaluating the integral from -2/5 to 2/5, we get:
= [ (25/3)(2/5)^3 - 4(2/5) ] - [ (25/3)(-2/5)^3 - 4(-2/5) ]
= (200/375) - (200/375)
= 0
Step 6: Set up the integral for the x-coordinate of the centroid (Cx):
Cx = (1/A) ∫[2/5,-2/5] x(25x^2 - 4) dx
Step 7: Evaluate the integral to find Cx:
Cx = (1/0) ∫[2/5,-2/5] x(25x^2 - 4) dx = undefined
Since the area of the region is 0, the x-coordinate of the centroid is undefined. This means that the centroid lies at the y-axis.
Therefore, the coordinates of the centroid for the region bounded by y = 25x^2 and y = 4 are (0, C), where C is the y-coordinate of the centroid.
To find the coordinates of the centroid of the region bounded by the curves y = 25x^2 and y = 4, we need to follow these steps:
1. Find the points of intersection between the two curves by setting them equal to each other:
25x^2 = 4
2. Solve the equation to find the x-values of the points of intersection. Taking the square root of both sides, we obtain:
x^2 = 4/25
x = ± 2/5
3. Now that we have the x-values of the points of intersection, we can find the y-values by substituting the x-values into either equation. Let's use the first equation, y = 25x^2:
y = 25(2/5)^2
y = 25(4/25)
y = 4
4. The points of intersection are (2/5, 4) and (-2/5, 4).
5. Next, we need to find the area of the region bounded by the curves. This can be done by finding the definite integral of the difference between the two functions:
Area = ∫[a,b] (upper curve - lower curve) dx
Where [a, b] represents the x-values of the points of intersection.
In this case, the difference between the curves is (25x^2 - 4). So the area becomes:
Area = ∫[-2/5, 2/5] (25x^2 - 4) dx
6. Evaluate the definite integral to find the area. A calculator or software can be used for this step. The result is the area of the region.
7. Finally, we calculate the coordinates of the centroid using the formulas:
x-coordinate of centroid = (1/Area) * ∫[a,b] x * (upper curve - lower curve) dx
y-coordinate of centroid = (1/Area) * ∫[a,b] 1/2 * (upper curve + lower curve) * (upper curve - lower curve) dx
Plug in the values of the upper and lower curves and evaluate the integrals using the area obtained in step 6.
The obtained (x, y) coordinates will represent the centroid of the region bounded by the curves y = 25x^2 and y = 4.