Question

Find the centroid (x¯,y¯) of the region bounded by y=6x²/7x, y=0, x=0, and x=6.

a) (9/5, 18/5)
b) (5/9, 5/18)
c) (3/2, 27/7)
d) (7/3, 7/27)

Answer 1

To find the centroid of the given region, we first simplify the function y = 16x²/7x to y = 6x, calculate the area, and then integrate to find the x and y coordinates of the centroid.

Step-by-step explanation:

To find the centroid (ׯ,y¯) of the region bounded by the given curves y = ¹⁶x²/7x, y = 0, x = 0, and x = 6, we first simplify the equation y = ¹⁶x²/7x to y = ⁶x by canceling the common factor x from the numerator and the denominator (assuming x ≠ 0). The boundaries of the region are then the x-axis (y = 0), the y-axis (x = 0), and the vertical line x = 6. To find the x-coordinate of the centroid, we use the formula ׯ = ¹/(A) ∫ (x)(y)dA, where A is the area of the region. To find the y-coordinate of the centroid, we use the formula y¯ = ¹/(2A) ∫ (y²)dA.

First, we calculate the area A of the region:

1. ׯ = ¹/(A) ∫ (x)(⁶x)dx from x = 0 to x = 6.
2. y¯ = ¹/(2A) ∫ (⁶x)dx from x = 0 to x = 6.

After performing the integration and simplifying, we will get specific numerical values for ׯ and y¯, which we can then match with the options provided to find the correct centroid coordinates.