Find the centroid $(\bar{x}, \bar{y})$ of the region bounded by: $y=5 x^2+6 x, \quad y=0, \quad$ and $\quad x=8$
Answer (1)
Calculate the area A of the region using the definite integral: A = − ∫ 0 8 ( 5 x 2 + 6 x ) d x = − 3 3136 .
Determine the x-coordinate of the centroid using the formula: x ˉ = A 1 ∫ 0 8 x ( − ( 5 x 2 + 6 x )) d x = 49 288 .
Determine the y-coordinate of the centroid using the formula: y ˉ = A 1 ∫ 0 8 2 1 ( − ( 5 x 2 + 6 x ) ) 2 d x = 49 5424 .
State the centroid coordinates: ( 49 288 , 49 5424 ) .
Explanation
Problem Introduction We are asked to find the centroid ( x ˉ , y ˉ ) of the region bounded by the curves y = 5 x 2 + 6 x , y = 0 , and x = 8 . The centroid represents the 'average' position of all the points in the region.
Finding the Area First, we need to find the area A of the region. The area is given by the integral of the function y = 5 x 2 + 6 x from x = 0 to x = 8 . Since the region is bounded by y = 0 below, the area is A = ∫ 0 8 ( 0 − ( 5 x 2 + 6 x )) d x = − ∫ 0 8 ( 5 x 2 + 6 x ) d x
Calculating the Area Let's calculate the integral: A = − ∫ 0 8 ( 5 x 2 + 6 x ) d x = − [ 3 5 x 3 + 3 x 2 ] 0 8 = − ( 3 5 ( 8 3 ) + 3 ( 8 2 ) ) = − ( 3 5 ( 512 ) + 3 ( 64 ) ) = − ( 3 2560 + 192 ) = − ( 3 2560 + 576 ) = − 3 3136 ≈ − 1045.33
Finding the x-coordinate of the centroid Next, we need to find the x-coordinate of the centroid, x ˉ . This is given by x ˉ = A 1 ∫ 0 8 x ( 0 − ( 5 x 2 + 6 x )) d x = A 1 ∫ 0 8 − ( 5 x 3 + 6 x 2 ) d x
Calculating the integral for x-coordinate Let's calculate the integral: ∫ 0 8 − ( 5 x 3 + 6 x 2 ) d x = − ∫ 0 8 ( 5 x 3 + 6 x 2 ) d x = − [ 4 5 x 4 + 2 x 3 ] 0 8 = − ( 4 5 ( 8 4 ) + 2 ( 8 3 ) ) = − ( 4 5 ( 4096 ) + 2 ( 512 ) ) = − ( 5120 + 1024 ) = − 6144
Calculating the x-coordinate Now, we can find x ˉ :
x ˉ = − 3 3136 − 6144 = 3 3136 6144 = 3136 6144 × 3 = 3136 18432 = 49 288 ≈ 5.877
Finding the y-coordinate of the centroid Next, we need to find the y-coordinate of the centroid, y ˉ . This is given by y ˉ = A 1 ∫ 0 8 2 1 ( 0 2 − ( 5 x 2 + 6 x ) 2 ) d x = A 1 ∫ 0 8 − 2 1 ( 25 x 4 + 60 x 3 + 36 x 2 ) d x
Calculating the integral for y-coordinate Let's calculate the integral: ∫ 0 8 − 2 1 ( 25 x 4 + 60 x 3 + 36 x 2 ) d x = − 2 1 ∫ 0 8 ( 25 x 4 + 60 x 3 + 36 x 2 ) d x = − 2 1 [ 5 x 5 + 15 x 4 + 12 x 3 ] 0 8 = − 2 1 ( 5 ( 8 5 ) + 15 ( 8 4 ) + 12 ( 8 3 ) ) = − 2 1 ( 5 ( 32768 ) + 15 ( 4096 ) + 12 ( 512 ) ) = − 2 1 ( 163840 + 61440 + 6144 ) = − 2 1 ( 231424 ) = − 115712
Calculating the y-coordinate Now, we can find y ˉ :
y ˉ = − 3 3136 − 115712 = 3 3136 115712 = 3136 115712 × 3 = 3136 347136 = 49 5424 ≈ 110.694
Final Answer Therefore, the centroid of the region is ( 49 288 , 49 5424 ) .
Examples
Imagine you are designing a solar panel with a curved surface to maximize sunlight capture. Knowing the centroid of the panel's area helps you determine the optimal location for mounting the panel to ensure balanced weight distribution and stability against wind. This ensures the structure is robust and efficient in harnessing solar energy.
