Find the centroid of the area bonded by the parabola y =4-x^2 and the x-axis?
A.(0,1.6) <<<< correct?
B.(0,1.7)
C.(0,1.8)
D.(0,1.9)
Let f(x)=4-x^2
f(x)=0 at x=±2
So integration is done from x=-2 to +2.
Note that dA = f(x)dx
Area, A
= ∫dA
= ∫f(x)dx
= ∫(4-x^2)dx
= [4x-x^3/3] x=-2 to +2
= [8-8/3 -(-8) -(-8/3)]
= 16-16/3
= 32/3
Moments about y-axis:
∫xdA
=∫x*f(x)dx
= 0 (by symmetry, or you can do the integration)
so x-centroid = 0
Moments about x-axis
∫(y/2)dA
=∫(f(x)/2)*f(x)dA
=256/15
So y-centroid
= (256/15) / A
= (256/15) / (32/3)
= 8/5
= 1.6
So the centroid is at (0,1.6), so A is correct.
To find the centroid of the area bounded by the parabola y = 4-x^2 and the x-axis, you need to calculate the coordinates (x, y) of the centroid. The centroid is the point (x, y) where the area is evenly distributed.
First, let's find the limits of integration. Since the parabola intersects the x-axis at the points (-2, 0) and (2, 0), the limits of integration for x will be -2 to 2.
Next, we need to find the area bounded by the parabola and the x-axis. This can be done using integration. The formula to calculate the area under a curve is given by:
Area = ∫[a,b] y dx
In this case, the formula becomes:
Area = ∫[-2,2] (4-x^2) dx
To calculate this integral, you can expand the expression and integrate each term separately:
Area = ∫[-2,2] 4 dx - ∫[-2,2] x^2 dx
Simplifying further:
Area = 4x|[-2,2] - (1/3)x^3|[-2,2]
Evaluating this expression:
Area = [4(2) - 4(-2)] - [(1/3)(2^3) - (1/3)(-2^3)]
Area = 16 - (8/3)
Area = 16/3
Now, to find the x-coordinate of the centroid, we need to calculate the first moment of area about the y-axis. The formula for the first moment of area is given by:
Mx = ∫[a,b] x(y) dx
In this case, the formula becomes:
Mx = ∫[-2,2] x(4-x^2) dx
Expanding the expression and integrating each term separately:
Mx = ∫[-2,2] (4x - x^3) dx
Simplifying further:
Mx = (4/2)x^2 - (1/4)x^4|[-2,2]
Evaluating this expression:
Mx = (4/2)(2^2) - (1/4)(2^4) - [(4/2)(-2^2) - (1/4)(-2^4)]
Mx = 8 - 4 - (-8 - 4)
Mx = 8 - 4 + 8 + 4
Mx = 16
Now, the x-coordinate of the centroid is given by:
xc = Mx / area
xc = 16 / (16/3)
xc = 3
Finally, to find the y-coordinate of the centroid, we can use the symmetry of the parabola. Since the curve is symmetric about the y-axis, the y-coordinate of the centroid will be the y-coordinate of the vertex of the parabola.
To find the vertex, we can use the formula x = -b / (2a). In this case, a = -1, b = 0, and c = 4. Thus, the x-coordinate of the vertex is x = -0 / (2*(-1)) = 0.
Substituting this x-coordinate into the equation of the parabola, we get:
y = 4 - 0^2 = 4
Therefore, the y-coordinate of the centroid is y = 4.
Hence, the coordinates of the centroid are (0, 4).
Comparing the calculated coordinates with the answer choices provided, the correct answer is A.(0,1.6) is incorrect, and the correct answer is not listed among the options provided.