Centroid - Area

Category: Physics, Mechanics

"Published in Newark, California, USA"

Find the centroid of the area for the given figure below

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

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

Photo by Math Principles in Everyday Life

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").

Simplifying Complex Fraction, 2

Category: Algebra

"Published in Newark, California, USA"

Simplify

Solution:

Consider the give equation above

Get the Least Common Denominator (LCD) at the numerator and denominator and then simplify, we have

Get the reciprocal of the divisor and perform the multiplication, we have

Simplify the above equation and therefore,

Integration - Algebraic Substitution

Category: Integral Calculus, Algebra

"Published in Suisun City, California, USA"

Evaluate

Solution:

Consider the given equation above

Since there's a radical function in the denominator that is included in the polynomial, we have to eliminate the radical function by algebraic substitution as follows

Let 

Substitute the above values to the given equation, we have

But

Therefore,