Find the centroid (x¯,y¯) of the region bounded by:
y=4x2+4x, y=0, x=0, and x=7.
To find the centroid (x¯,y¯) of the region bounded by the given equations, we need to compute the coordinates of the centroid using the following formulas:
x¯ = (1/A) ∫ x dA
y¯ = (1/A) ∫ y dA
Where A is the area of the region, and the integrals are taken over the region.
First, we need to find the area, A, of the region. The region is bounded by the x-axis (y = 0), y = 4x^2 + 4x, x = 0, and x = 7. To find the area, we integrate the region from x = 0 to x = 7:
A = ∫[0 to 7] (4x^2 + 4x) dx
Let's calculate this integral:
A = ∫[0 to 7] (4x^2 + 4x) dx
= ∫[0 to 7] 4x^2 dx + ∫[0 to 7] 4x dx
Integrating term by term:
= [4/3 x^3] | [0 to 7] + [2x^2] | [0 to 7]
= (4/3 * 7^3) + (2 * 7^2)
= (4/3 * 343) + (2 * 49)
= 1372/3 + 98
= 1372/3 + 294/3
= 1666/3
= 555.33
So, the area of the region is A = 555.33.
Next, we calculate the x-coordinate of the centroid, x¯:
x¯ = (1/A) ∫ x dA
To calculate this, we need to find ∫ x dA. Since x does not depend on y, the integral becomes:
∫ x dA = x ∫ dA
The integral of dA is just the area of the region, A. Therefore, the integral becomes:
∫ x dA = x A
Substituting A = 555.33, we have:
x¯ = (1/555.33) ∫ x dA
= (1/555.33) (x A)
= (1/555.33) (x * 555.33)
= x
So, the x-coordinate of the centroid is simply the average value of x over the region.
Lastly, we compute the y-coordinate of the centroid, y¯:
y¯ = (1/A) ∫ y dA
To calculate this, we need to find ∫ y dA. Since y is a function of x, we need to express y in terms of x using the equation y = 4x^2 + 4x.
Substituting this equation into the integral, we have:
y¯ = (1/A) x ∫ (4x^2 + 4x) dx
Integrating term by term:
y¯ = (1/A) x * [(4/3 x^3) + (2 x^2)]
= x * [(4/3 x^3) + (2 x^2)] / (555.33)
We can simplify this expression, but it will be quite complicated. Since the expression is already in terms of x, let's find the average value of y over the region by evaluating y at the two boundaries:
y¯ = (1/A) x [(4/3 x^3) + (2 x^2)] / (555.33)
= [(4/3 x^3) + (2 x^2)] / 555.33
Evaluating this expression at the boundaries x = 0 and x = 7, we have:
y¯ = [(4/3 * 7^3) + (2 * 7^2)] / 555.33
= (4/3 * 343) + (2 * 49) / 555.33
= (544/3 + 98) / 555.33
= 1832/3 / 555.33
= 610.67 / 555.33
≈ 1.10
So, the y-coordinate of the centroid is y¯ ≈ 1.10.
Therefore, the centroid (x¯,y¯) of the region bounded by the equations y = 4x^2 + 4x, y = 0, x = 0, and x = 7 is approximately (x¯,y¯) = (x, 1.10). Since x is not specified, the x-coordinate of the centroid can be any value within the region.
Of course the x value can be determined.
x¯ = (1/A) ∫ x dA = (1/A) ∫ x f(x) dx
y¯ = (1/A) ∫ y dA = (1/A) ∫ y f(x) dx
Apologies for the confusion. Let's proceed to calculate the x-coordinate of the centroid.
To find x¯, we use the formula:
x¯ = (1/A) ∫ x f(x) dx
In this case, f(x) is represented by the equation y = 4x^2 + 4x. Therefore, we have:
x¯ = (1/A) ∫ x (4x^2 + 4x) dx
Simplifying the integral:
x¯ = (1/A) ∫ (4x^3 + 4x^2) dx
= (1/A) (4/4 x^4 + 4/3 x^3) + C
= (1/A) (x^4 + 4/3 x^3) + C
Now, we need to evaluate this integral from x = 0 to x = 7:
x¯ = (1/A) [(7^4 + 4/3 * 7^3) - (0^4 + 4/3 * 0^3)]
= (1/A) [(2401 + 2156/3) - (0 + 0)]
= (1/A) (2401 + 2156/3)
= (1/A) (7203/3 + 2156/3)
= (1/A) (9359/3)
Since we found earlier that A = 555.33, we substitute this value:
x¯ = (1/555.33) (9359/3)
= 9359/16659.99
≈ 0.561
Therefore, the x-coordinate of the centroid (x¯) is approximately 0.561.
Now, let's calculate the y-coordinate of the centroid.
To find y¯, we use the formula:
y¯ = (1/A) ∫ y f(x) dx
Substituting the equation y = 4x^2 + 4x into the formula, we have:
y¯ = (1/A) ∫ (4x^2 + 4x)(4x^2 + 4x) dx
= (1/A) ∫ (16x^4 + 32x^3 + 16x^2) dx
= (1/A) (16/5 x^5 + 8 x^4 + 16/3 x^3) + C
= (1/A) (8/10 x^5 + 24/15 x^4 + 16/15 x^3) + C
= (1/A) (4/5 x^5 + 8/15 x^4 + 4/15 x^3) + C
Now, we evaluate this integral from x = 0 to x = 7:
y¯ = (1/A) [(4/5 * 7^5 + 8/15 * 7^4 + 4/15 * 7^3) - (0^5 + 0^4 + 0^3)]
= (1/A) [(4/5 * 16807 + 8/15 * 2401 + 4/15 * 343) - 0]
≈ (1/A) (13445.03)
Substituting the value of A = 555.33:
y¯ ≈ (1/555.33) (13445.03)
≈ 24.19
Therefore, the y-coordinate of the centroid (y¯) is approximately 24.19.
Hence, the centroid (x¯, y¯) of the region bounded by the equations y = 4x^2 + 4x, y = 0, x = 0, and x = 7 is approximately (0.561, 24.19).
To find the centroid of a region bounded by a set of curves, we need to compute the moments of the region with respect to the x and y axes. The centroid coordinates (x¯, y¯) can then be calculated using the following formulas:
x¯ = M_y / A
y¯ = M_x / A
where:
- M_y is the moment of the region with respect to the y-axis
- M_x is the moment of the region with respect to the x-axis
- A is the area of the region
Let's use these formulas to find the centroid of the given region:
To find the moments M_y and M_x, we need to evaluate double integrals. However, since the region is specified by algebraic curves, we can instead make use of single integrals for our calculations.
The equation y = 4x^2 + 4x is the top boundary of the region, while y = 0 is the bottom boundary. The region extends from x = 0 to x = 7.
First, let's find the x-coordinate of the centroid, x¯.
To calculate the moment with respect to the y-axis (M_y), we integrate the product of the distance from the y-axis squared (x^2) and the differential area (dA) along the y-direction:
M_y = ∫(x^2 * dA)
Since the region is bounded by y = 4x^2 + 4x and y = 0, we can rewrite the integral as:
M_y = ∫[0, 7] ∫[0, 4x^2 + 4x] (x^2 * dy) dx
Let's solve this integral step-by-step:
M_y = ∫[0, 7] (x^2 * (4x^2 + 4x)) dx
= ∫[0, 7] (4x^4 + 4x^3) dx
= (4/5)x^5 + (4/4)x^4 | [0, 7]
= (4/5)(7^5) + (4/4)(7^4) - (4/5)(0) - (4/4)(0)
= (4/5)(16807) + (4/4)(2401)
= 13445.6 + 2401
= 15846.6
Next, let's calculate the area of the region. Since we are given the equations of the boundaries, we can determine the area using a single integral:
A = ∫[0, 7] (4x^2 + 4x) dx
Let's compute this integral step-by-step:
A = ∫[0, 7] (4x^2 + 4x) dx
= (4/3)x^3 + (4/2)x^2 | [0, 7]
= (4/3)(7^3) + (4/2)(7^2) - (4/3)(0) - (4/2)(0)
= (4/3)(343) + (4/2)(49)
= 137.333 + 98
= 235.333
Now, we have both M_y = 15846.6 and A = 235.333. Let's calculate x¯ using the formula:
x¯ = M_y / A = 15846.6 / 235.333
≈ 67.37
Therefore, the x-coordinate of the centroid is approximately 67.37.
To find the y-coordinate of the centroid, y¯, we need to calculate the moment with respect to the x-axis (M_x).
M_x = ∫(y * dA)
Since the region is bounded by y = 4x^2 + 4x and y = 0, we can rewrite the integral as:
M_x = ∫[0, 7] ∫[0, 4x^2 + 4x] (y * dy) dx
Let's solve this integral step-by-step:
M_x = ∫[0, 7] (y^2 / 2) | [0, 4x^2 + 4x] dx
= ∫[0, 7] ((4x^2 + 4x)^2 / 2) dx
= ∫[0, 7] (8x^4 + 16x^3 + 8x^2) dx
= (1/5)x^5 + (4/4)x^4 + (8/3)x^3 | [0, 7]
= (1/5)(7^5) + (4/4)(7^4) + (8/3)(7^3) - (1/5)(0) - (4/4)(0) - (8/3)(0)
= (1/5)(16807) + (4/4)(2401) + (8/3)(343)
= 13445.6 + 2401 + 960.888
= 16707.488
Now, we have M_x = 16707.488. To find y¯, we use the formula:
y¯ = M_x / A = 16707.488 / 235.333
≈ 70.987
Therefore, the y-coordinate of the centroid is approximately 70.987.
Hence, the centroid of the region bounded by y = 4x^2 + 4x, y = 0, x = 0, and x = 7 is (x¯, y¯) ≈ (67.37, 70.987).