Problem 24 - Single Variable Calculus

Question

Find the centroid of the region bounded by the graphs of the given equations. $$ y=-x^{2}+3, \quad y=x^{2}-2 x-1 $$

Step-by-Step Solution

Answer

The intersection points of the given equations are at \(x=-1\) and \(x=2\). The area of the bounded region is \(A=\frac{27}{2}\). The centroid coordinates can be calculated using the formulas, and we get \(x_{c}=\frac{1}{2}\), \(y_{c}=\frac{7}{6}\). Therefore, the centroid of the region is at \((\frac{1}{2}, \frac{7}{6})\).

Step 1: 1. Find intersection points of the graphs

To determine the intersection points, we must find where the two functions are equal, i.e., we need to solve the equation: \(-x^{2}+3=x^{2}-2x-1\). By solving this equation, we can find the x-coordinates of the intersection points.

Step 2: 2. Determine the integral of the area of the bounded region

To find the area of the bounded region, we'll need to integrate the difference of the two functions along the x-axis from the lower intersection point to the upper intersection point. The area can be calculated using the formula: \(A=\int_{a}^{b}(f(x)-g(x))dx\), where f(x) is the function with the higher value in the given interval, and g(x) is the function with the lower value. This will give us the area of the region which is required to calculate the centroid coordinates later.

Step 3: 3. Find the centroid coordinates

To find the centroid of the region, we will use the centroid formula for a plane region. The centroid coordinates are: \(x_{c}=\frac{1}{A}\int_{a}^{b}x(f(x)-g(x))dx\) \(y_{c}=\frac{1}{A}\int_{a}^{b}\frac{f(x)+g(x)}{2}(f(x)-g(x))dx\) Where A is the area of the region, f(x) and g(x) are the given functions, and a and b are the x-coordinates of the intersection points. Calculate the centroid coordinates using these formulas.

Key Concepts

Intersection Points

Understand intersection points by finding where the two functions intersect each other. For the functions provided, the intersection point is where both equations provide the same value of \(y\) for a specific \(x\). In this problem, we solve

• \(-x^{2} + 3 = x^{2} - 2x - 1\)

By solving this, we'll find values of \(x\) where the two graphs meet. These values are critical as they form the limits of integration for calculating area and later the centroid coordinates.
Finding intersection points might be one of the first steps in solving boundary problems. They help confirm how graphs overlap or meet on a plane.

Integral of the Area

The integral of an area between two curves is essential to find how much space is enclosed by these curves. To calculate this, you take the definite integral of the difference between the two functions.

• Identify which function forms the top curve and which the bottom one within the intersection interval.
• Subtract the lower function from the upper function to get the area function.
• Integrate this difference over the interval determined by the intersection points.

In our case:\[ A = \int_{a}^{b} (-x^2 + 3 - (x^2 - 2x - 1)) \, dx \]Where \(a\) and \(b\) are the intersection points. The result gives you a scalar value, crucial for further centroid calculation.

Centroid Coordinates

The centroid is like the balancing point of the region, calculated using averages of coordinate positions weighted by area. To find the centroid's coordinates \((x_c, y_c)\):

• Use the formula for \(x_c\), which involves the integral of \(x\) times the area function, divided by the total area \(A\).
• Use the formula for \(y_c\), a weighted average around the y-axis dependent on the vertical height of the centroid at any point, also divided by \(A\).

Equations for computing the centroid:\[ x_c = \frac{1}{A} \int_{a}^{b} x(f(x)-g(x)) \, dx \]\[ y_c = \frac{1}{A} \int_{a}^{b} \frac{f(x) + g(x)}{2} (f(x) - g(x)) \, dx \]These provide the navigational "compass points" for the geometric region.

Calculus Integration

Calculus integration in the context of physics and geometry involves determining quantities like area and centroids. When we integrate a function over an interval, we calculate the net accumulation of the function's value across that interval. For a region between two curves:

• Start by establishing the bounds of integration via intersection points.
• Apply definite integration to find the net difference between the curves, giving area.
• Use integration for centroid calculations to evaluate "mass concentration" over this area.

Integration provides the means to capture the behavior of changing variables, such as areas under curves or volumetric balance points.