Centroid - Area, 2

Category: Integral Calculus, Analytic Geometry, Algebra, Physics, Mechanics

"Published in Newark, California, USA"

Find the centroid of the area bounded by two curves for

Solution:

The first thing that we have to do is to draw or sketch the two given curves using the principles of Analytic Geometry as follows

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Next, we need to get their points of intersection by solving the two equations, two unknowns as follows

but

The above equation becomes

Equate each factor to zero and solve for the value of y. Therefore, y = 2 and y = -1.

Substitute the values of y either of the two equation in order to solve for the value of x, we have

If y = 2, then

If y = -1, then

Their points of intersection are (1, -1) and (4, 2).

Label further the figure and draw the horizontal strip, we have

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The area bounded by the two curves is 

The x value of the centroid for the figure bounded by two curves is given by the formula

If the length of a strip is x, then x_C is ½ x. The above equation becomes

Therefore,

The y value of the centroid for the figure bounded by two curves is given by the formula

If the length of a strip is x, then y_C is also equal to y which is the distance of a strip from x axis. Since dy is a very small measurement, then dy is negligible. The above equation becomes

Therefore,

Therefore, the coordinates of the centroid are 

Centroid - Area

Category: Physics, Mechanics

"Published in Newark, California, USA"

Find the centroid of the area for the given figure below

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Solution:

Consider the given figure above

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Put the given figure above in x and y axis and assign the origin at the lower left corner of the figure. Divide the given figure into three rectangles and assign their center. Label further the given figure above, we have

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Consider A₁:

Area A₁ = LW = (5 in)(2 in) = 10 in²
x₁ = ½(5) = 2.5
y₁ = ½(2) + 7 = 8
Therefore, C₁(2.5, 8)

Consider A₂:

Area A₂ = LW = (3 in)(4 in) = 12 in²
x₂ = ½(3) = 1.5
y₂ = ½(4) + 3 = 5
Therefore, C₂(1.5, 5)


Consider A₃:

Area A₃ = LW = (6 in)(3 in) = 18 in²
x₃ = ½(6) = 3
y₃ = ½(3) = 1.5
Therefore, C₃(3, 1.5)

Area A_T = A₁ + A₂ + A₃
             = 10 in² + 12 in² + 18 in²
             = 40 in²

Now, we can get the coordinates of the Centroid of the given figure. Let's get the x value of the Centroid as follows

Let's get the y value of the Centroid as follows

Therefore, the coordinates of the Centroid is (2.425", 4.175").