Find the centroid of the area bounded by the parabola y=4-x^2 and the x-axis
A.(0,1.6)
B.(0,1.7)
C.(0,1.8)
D.(0,1.9)
domain from -2 to +2 but due to symmetry just do from 0 to 2
Area = int y dx = 4 x-x^3/3 from 0 to 2
= 8 -8/3 = 16/3
moment = (1/2)int y^2 dx
= (1/2) int[ 16-8x^2+x^4]
= (1/2) [ 16 x - 8 x^3/3 + x^5/5] 0 to 2
= (1/2)[32 -64/3 +32/5]
= 8.5333
moment/area = 8.5333*3/16
= 1.6
another senior moment for me.
Why did the parabola go to therapy? It had too many axes to grind!
The centroid of the area bounded by the parabola y=4-x^2 and the x-axis is located at (0, 1.6). So, the answer is A. (0, 1.6). Keep calm and find the centroid!
To find the centroid of the area bounded by the parabola y=4-x^2 and the x-axis, we need to compute the coordinates of the centroid using integration.
First, let's find the points of intersection between the parabola and the x-axis. Setting y=0, we solve the equation 4-x^2=0 for x:
4 - x^2 = 0
x^2 = 4
x = ±√4
x = ±2
So, the parabola intersects the x-axis at x = -2 and x = 2.
To find the x-coordinate of the centroid, we integrate x * f(x) over the interval [-2, 2], where f(x) is the equation of the parabola y=4-x^2:
x-coordinate of centroid (Xₒ) = ∫(x * f(x)) dx over [-2, 2]
= ∫(x * (4-x^2)) dx over [-2, 2]
Now, integrate the expression (x * (4-x^2)) with respect to x:
Using the Power Rule of Integration:
∫(x * (4-x^2)) dx = 4∫(x) dx - ∫(x^3) dx
= 4 * (x^2/2) - (x^4/4) + C
Evaluate the expression at the upper and lower limits of integration:
= [4 * (x^2/2) - (x^4/4)] evaluated at [2, -2]
= [4 * (2^2/2) - (2^4/4)] - [4 * (-2^2/2) - (-2^4/4)]
= [4 * 2 - 4] - [4 * 2 - 4]
= (8 - 4) - (8 - 4)
= 4 - 4
= 0
So, the x-coordinate of the centroid (Xₒ) is 0.
To find the y-coordinate of the centroid, we integrate f(x) dx over the interval [-2, 2]:
y-coordinate of centroid (Yₒ) = ∫(f(x)) dx over [-2, 2]
= ∫(4-x^2) dx over [-2, 2]
Now, integrate the expression (4-x^2) with respect to x:
∫(4-x^2) dx = 4∫(1) dx - ∫(x^2) dx
= 4 * x - (x^3/3) + C
Evaluate the expression at the upper and lower limits of integration:
= [4 * x - (x^3/3)] evaluated at [2, -2]
= [4 * 2 - (2^3/3)] - [4 * -2 - (-2^3/3)]
= [8 - 8/3] - [-8 + 8/3]
= 24/3 - 24/3
= 0
So, the y-coordinate of the centroid (Yₒ) is 0.
Therefore, the coordinates of the centroid are (0,0).
None of the given options A, B, C, or D correspond to the correct answer.
In general, finding the centroid of a curved region can be a very messy question.
See this pdf page, with 2 examples at the beginning
http://pages.pacificcoast.net/~cazelais/187/centroids.pdf
Because of the symmetry of your equation, we know that the centroid has to be on the y-axis, as seen by your choices of answers.
It must be up the y-axis in such a way that the area above must be equal to the area below, or 1/2 the total area
total area = 2∫(4-x^2 ) dx from 0 to 2
= 2[ 4x - x^3/3] from 0 to 2
= 2[ 8 - 8/3 - 0] = 32/3
so 1/2 the area is 16/3
taking horizontal slices from y to 4
area = ∫(4-y)^.5 dy from y to 4
= [ (-2/3)(4-y)^(3/2) ] from y to 4
= [ (-2/3)(0)^(3/2) - (-2/3)(4-y)^(3/2) ] = 8/3 only considering the area in 1st quadrant
(2/3)(4-y)^(3/2) = 8/3
(4-y)^3/2 = 4
4 - y = 4^(2/3) = 2.5198
y = 1.48
ARGGG! , none of your choices, check my arithmetic