Centroid of lamina w/ eqn y = x(3-x), starting at x = 0.5, 1.5 wide

[Joeyboiiiiii]

2. The coordinates for the centroid of a lamana is calculated using the summation formulas given below:

. . . . .\(\displaystyle \bar{x}\, =\, \dfrac{\sum_{x=a}^{x=b}x\delta A}{\sum_a^b \delta A}\)

. . . . .\(\displaystyle \bar{y}\, =\, \dfrac{\sum_{y=a}^{y=b}y\delta A}{\sum_a^b \delta A}\)

a. Derive, from first principles, formulas for \(\displaystyle \bar{x}\) and \(\displaystyle \bar{y}\) involving integrals.

b. A Guillotine blade in the form of a lamina is shown in the graphic. Use your derived formulas for \(\displaystyle \bar{x}\) and \(\displaystyle \bar{y}\) to calculate its centroid. (The curve has equation \(\displaystyle y\, =\, x\, (3\, -\, x).\) The left-hand end is at \(\displaystyle x\, =\, 0.5.\))

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This question has has me stuck for about three weeks now

 

[tkhunny]

It makes no sense to be so stuck. Why have you been given no tools to sole the problem you have been given.

You need three integrals. Please calculate the area of the entire figure - using an integral. This is the first step. This is the denominator for both fractions.

Go!

 

[stapel]

2. The coordinates for the centroid of a lamana is calculated using the summation formulas given below:

. . . . .\(\displaystyle \bar{x}\, =\, \dfrac{\sum_{x=a}^{x=b}x\delta A}{\sum_a^b \delta A}\)

. . . . .\(\displaystyle \bar{y}\, =\, \dfrac{\sum_{y=a}^{y=b}y\delta A}{\sum_a^b \delta A}\)

a. Derive, from first principles, formulas for \(\displaystyle \bar{x}\) and \(\displaystyle \bar{y}\) involving integrals.

b. A Guillotine blade in the form of a lamina is shown in the graphic. Use your derived formulas for \(\displaystyle \bar{x}\) and \(\displaystyle \bar{y}\) to calculate its centroid. (The curve has equation \(\displaystyle y\, =\, x\, (3\, -\, x).\) The left-hand end is at \(\displaystyle x\, =\, 0.5.\))

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View attachment 8969 This question has has me stuck for about three weeks now

What are "a" and "b"? Are they, for the second part of the question, equal to x = 0.5 and x = 2? What is "A"? What is your understanding of "delta-A"? What is "x" in the summations? What "first principles" have you been given?

Thank you!