91Ó°ÊÓ

To find the centroid of a region, we must first identify the region's shape. The region in question is defined by the curves and lines: \( y = x^3 \), \( y = 0 \), and \( x = 1 \). Imagining this on the Cartesian plane:

• \( y = x^3 \) is a curve starting from the origin and curving upward as \( x \) increases.
• \( y = 0 \) represents the x-axis, which acts as a lower boundary.
• Finally, \( x = 1 \) is a vertical line, marking the end of our region horizontally.

The region is therefore above the x-axis, under the curve \( y = x^3 \), and bounded on the right by \( x=1 \). This creates a region whose horizontal span is from \( x=0 \) to \( x=1 \). Understanding the bounds is essential, as they define the space within which we calculate the centroid.

Integration helps us find areas and averages in calculus, which are crucial in determining centroids. The area \( A \) of our region is found by integrating the function \( y = x^3 \) from \( x=0 \) to \( x=1 \). This is done as follows:\[ A = \int_{0}^{1} x^3 \, dx \]The process of integration gives us the accumulated value of \( x^3 \) over the interval, found to be \( \frac{1}{4} \). Finding this area is the first step toward identifying the centroid because it will be used to calculate the average positions of \( x \) and \( y \). Integration is also employed to compute the weighted sums of positions—these later steps are crucial for determining the x-coordinate and y-coordinate of the centroid. Thus, integration is more than just finding areas; it's about utilizing these calculations for positional averages.

The centroid (\( \bar{x}, \bar{y} \)) represents the geometric center of the region. We compute this using weighted averages of the x and y positions. To find the x-coordinate \( \bar{x} \): - We use the weighted integral of \( x \) times the function: \[ \bar{x} = \frac{1}{A} \int_{0}^{1} x \cdot x^3 \, dx = \frac{4}{5} \]This indicates that the balance point along the x-axis happens around \( 0.8 \).For the y-coordinate \( \bar{y} \):- The formula considers the average value of the function, \[ \bar{y} = \frac{1}{A} \int_{0}^{1} \frac{x^3}{2} x^3 \, dx = \frac{2}{7} \]This signifies the average y-position or height of the region is approximately \(0.286\).These calculations secure the coordinates of the centroid as \( \left( \frac{4}{5}, \frac{2}{7} \right) \), which centers our region both horizontally and vertically, fulfilling the requirement of finding where the region naturally balances.