Question

As needed, use a computer to plot graphs and to check values of integrals. Find the centroid of the area above \(y=x^{2}\) and below \(y=c(c > 0)\).

Short answer

The centroid is \( (0, \frac{3c}{5}) \)

Step-by-step solution

- Understand the Region

Identify the region bounded by the curves. The region of interest is the area above the parabola \(y=x^{2}\) and below the horizontal line \(y=c\) (where \(c>0\)).

- Find Intersection Points

Solve for the intersection points of the curves \(y=x^{2}\) and \(y=c\). Set \[x^{2}=c\] and solve for \(x\). The intersection points are \[x=-\sqrt{c}\] and \[x=\sqrt{c}\].

- Area of the Region

Find the area of the region bounded by the two curves. The area \(A\) can be found by integrating the difference of the functions: \[A = \int_{-\sqrt{c}}^{\sqrt{c}} (c - x^{2}) \, dx\].

- Calculate the Area

Compute the integral to find the area: \[A = \int_{-\sqrt{c}}^{\sqrt{c}} c \, dx - \int_{-\sqrt{c}}^{\sqrt{c}} x^{2} \, dx\]. This simplifies to \[A = 2c\sqrt{c} - \frac{2c^{3/2}}{3} = \frac{4c^{3/2}}{3}\].

- Find the Centroid Coordinates

The x-coordinate of the centroid \(\bar{x}\) is given by \[\bar{x} = \frac{1}{A} \int_{-\sqrt{c}}^{\sqrt{c}} x(c - x^{2}) \, dx\]. Due to symmetry, \[\bar{x} = 0\]. The y-coordinate of the centroid \(\bar{y}\) is given by \[\bar{y} = \frac{1}{A} \int_{-\sqrt{c}}^{\sqrt{c}} (c - x^{2})^{2} \, dx\].

- Calculate the Centroid’s Y-Coordinate

Compute the integral for \(\bar{y}\): \[\bar{y} = \frac{1}{A} \int_{-\sqrt{c}}^{\sqrt{c}} (c - x^{2})x^{2} \, dx\]. This simplifies to \[\bar{y} = \frac{1}{\frac{4c^{3/2}}{3}} \left(\frac{4c^{3/2}}{5}\right) = \frac{3c}{5}\].

Key concepts

Integration

Integration is a crucial mathematical tool used to find areas under curves and other relevant properties. In this exercise, we use integration to determine the area between the parabola and the horizontal line. This involves setting up an integral that represents the total area above the parabola and below the line.
We specifically use definite integration, meaning we calculate the integral from one boundary to another, which in this case are the intersection points.
Remember, integration is essentially the process of summing infinitely small pieces to find a whole.
Here, it helps us in accurately determining the area needed to locate the centroid.

Intersection Points

Intersection points are where two functions meet on a graph. Finding these points is crucial because they define the boundaries for integration.
In this problem, we identify the intersection points of the parabola, \(y = x^2\), and the horizontal line, \(y = c\).
Setting these equations equal, \(x^2 = c\), we solve for \(x\), finding \(x = -\forall c\) and \(x = \forall c\).
These points, \((-\forall c, c)\) and \((\forall c, c)\), form the limits of integration for calculating the area and centroids.

Area Calculation

Calculating the area of a region involves integrating the difference between the upper and lower functions within the determined boundaries.
In this exercise, the upper function is \(y = c\) and the lower function is \(y = x^2\).
The area \(A\) is found by integrating \((c - x^2)\) from \(x = -\forall c\) to \(x = \forall c\).
Splitting this into simpler integrals: \(A = \forall (c\forall) - 2\forall \frac{c^{3/2}}{3} = \frac{4c^{3/2}}{3}\).
This gives us the total area, necessary for calculating centroid coordinates.

Symmetry in Functions

Symmetry simplifies many mathematical problems, particularly those involving centroids.
A function is symmetric if it looks the same on both sides of a central axis. In this problem, the parabola \(y = x^2\) is symmetric about the y-axis, meaning that the area above it is mirrored evenly on both sides.
Because of this symmetry, we know that the x-coordinate of the centroid \(\forall x\) is zero, as the centroid lies directly on the y-axis.
Using symmetry often reduces complexity and aids in direct computational solutions, like determining the centroid in this exercise.