Find the centroid $(\bar{x}, \bar{y})$ of the region bounded by: $y=5 x^2+6 x, \quad y=0, \quad$ and $\quad x=8$
$\bar{x}=$ $\square$
$\bar{y}=$ $\square$
Answer (1)
Calculate the area A of the region using the integral: A = − ∫ 0 8 ( 5 x 2 + 6 x ) d x = 3 3136 .
Calculate 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 .
Calculate the y-coordinate of the centroid using the formula: y ˉ = 2 A 1 ∫ 0 8 ( 5 x 2 + 6 x ) 2 d x = 49 5424 .
State the centroid coordinates: ( x ˉ , y ˉ ) = ( 49 288 , 49 5424 ) .
Explanation
Problem Analysis We are given the equations y = 5 x 2 + 6 x , y = 0 , and x = 8 . We need to find the centroid ( x ˉ , y ˉ ) of the region bounded by these equations. The centroid represents the "average" position of all the points in the region.
Area Calculation First, we need to find the area of the region. The area A 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 and y = 5 x 2 + 6 x , and 5 x 2 + 6 x is negative on the interval [ 0 , 8 ] , we need to take the absolute value of the integral, which is equivalent to taking the negative of the integral. Thus, we have: A = − ∫ 0 8 ( 5 x 2 + 6 x ) d x
Calculating the Area Now, 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 Since area must be positive, we take the absolute value: A = 3 3136
Calculating x-coordinate of Centroid Next, we need to find x ˉ . The formula for x ˉ is: x ˉ = A 1 ∫ 0 8 x ( − 5 x 2 − 6 x ) d x = A 1 ∫ 0 8 ( − 5 x 3 − 6 x 2 ) d x Let's calculate the integral: ∫ 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 ) = − 5 ( 1024 ) − 1024 = − 5120 − 1024 = − 6144
Calculating x-coordinate of Centroid (Corrected) Now, we can find x ˉ :
x ˉ = A 1 ( − 6144 ) = 3136 3 ( − 6144 ) = 3136 − 18432 = − 196 1152 = − 98 576 = − 49 288 Since we are considering the area to be positive, we should have: x ˉ = A 1 ∫ 0 8 x ( 5 x 2 + 6 x ) d x = A 1 ∫ 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 ) = 5 ( 1024 ) + 1024 = 5120 + 1024 = 6144 x ˉ = A 1 ( 6144 ) = 3136 3 ( 6144 ) = 3136 18432 = 196 1152 = 98 576 = 49 288 ≈ 5.87755
Calculating y-coordinate of Centroid Next, we need to find y ˉ . The formula for y ˉ is: y ˉ = A 1 ∫ 0 8 2 1 ( 5 x 2 + 6 x ) 2 d x = 2 A 1 ∫ 0 8 ( 25 x 4 + 60 x 3 + 36 x 2 ) d x Let's calculate the integral: ∫ 0 8 ( 25 x 4 + 60 x 3 + 36 x 2 ) d x = [ 5 x 5 + 15 x 4 + 12 x 3 ] 0 8 = 5 ( 8 5 ) + 15 ( 8 4 ) + 12 ( 8 3 ) = 5 ( 32768 ) + 15 ( 4096 ) + 12 ( 512 ) = 163840 + 61440 + 6144 = 231424
Calculating y-coordinate of Centroid Now, we can find y ˉ :
y ˉ = 2 A 1 ( 231424 ) = 2 ( 3136 ) 3 ( 231424 ) = 6272 694272 = 392 43392 = 196 21696 = 98 10848 = 49 5424 ≈ 110.69387
Final Answer Therefore, the centroid is ( x ˉ , y ˉ ) = ( 49 288 , 49 5424 ) ≈ ( 5.87755 , 110.69387 ) .
Examples
Understanding centroids is very useful in various engineering applications. For example, when designing a bridge, engineers need to calculate the centroid of the bridge's cross-section to ensure stability and balance. Similarly, in architecture, knowing the centroid of a building's structure helps in distributing weight evenly and preventing structural failure. In sports, understanding the center of mass (similar to centroid) of an object like a baseball bat helps in optimizing its swing and performance.
