Problem 279 - Calculus Volume 1

Question

For the following exercises, use a calculator to draw the region, then compute the center of mass \((\overline{x}, \overline{y}) .\) Use symmetry to help locate the center of mass whenever possible. The region bounded by \(y=x^{2}\) and \(y=x^{4}\) in the first quadrant

Step-by-Step Solution

Answer

The center of mass is \(\left(\frac{5}{8}, \frac{1}{3}\right)\)."

Step 1: Identify the Region

First, identify the intersection points of the curves by setting the equations equal to each other: \(x^2 = x^4\). Solving for \(x\), we get \(x^2(x^2 - 1) = 0\), which results in \(x = 0\) or \(x = 1\). This means the region bounded by these curves in the first quadrant is from \(x=0\) to \(x=1\).

Step 2: Set Up Integral for Area

The area \(A\) of the region can be found by the integral of the difference between the top function \(y = x^2\) and the bottom function \(y = x^4\) within the interval [0,1]. The area integral is \( \int_0^1 (x^2 - x^4) \, dx\).

Step 3: Integrate to Find the Area

Compute the integral \( \int_0^1 (x^2 - x^4) \, dx = \left[ \frac{x^3}{3} - \frac{x^5}{5} \right]_0^1 = \frac{1}{3} - \frac{1}{5} = \frac{2}{15}\). This is the area of the region.

Step 4: Compute \(\overline{x}\) (x-coordinate of Center of Mass)

Use the formula for \(\overline{x}\): \( \overline{x} = \frac{1}{A} \int_a^b x(f(x) - g(x)) \, dx \) where \(f(x) = x^2\) and \(g(x) = x^4\). So, \( \overline{x} = \frac{15}{2} \int_0^1 x(x^2 - x^4) \, dx = \frac{15}{2} \int_0^1 (x^3 - x^5) \, dx\).

Step 5: Evaluate \(\overline{x}\) Integral

Compute \( \int_0^1 (x^3 - x^5) \, dx = \left[ \frac{x^4}{4} - \frac{x^6}{6} \right]_0^1 = \frac{1}{4} - \frac{1}{6} = \frac{1}{12} \). Therefore, \(\overline{x} = \frac{15}{2} \cdot \frac{1}{12} = \frac{5}{8}\).

Step 6: Compute \(\overline{y}\) (y-coordinate of Center of Mass)

Use the formula for \(\overline{y}\): \( \overline{y} = \frac{1}{A} \int_a^b \frac{(f(x)^2 - g(x)^2)}{2} \, dx \). So, \( \overline{y} = \frac{15}{2} \int_0^1 \frac{(x^4 - x^8)}{2} \, dx = \frac{15}{4} \int_0^1 (x^4 - x^8) \, dx\).

Step 7: Evaluate \(\overline{y}\) Integral

Compute \( \int_0^1 (x^4 - x^8) \, dx = \left[ \frac{x^5}{5} - \frac{x^9}{9} \right]_0^1 = \frac{1}{5} - \frac{1}{9} = \frac{4}{45} \). Therefore, \(\overline{y} = \frac{15}{4} \cdot \frac{4}{45} = \frac{1}{3}\).

Step 8: Final Step: Conclusion

The center of mass of the region bounded by \(y = x^2\) and \(y = x^4\) in the first quadrant is \(\left(\overline{x}, \overline{y}\right) = \left(\frac{5}{8}, \frac{1}{3}\right)\).

Key Concepts

Integration Basics

Integration is a core concept in calculus that allows us to find areas under curves, among other applications. By integrating, we can calculate the total accumulated quantity, such as area, displacement, or even mass when dealing with physical contexts.

In the context of finding the center of mass of a region, integration helps us aggregate small pieces of the area and consider their contribution relative to their position.

• To set up an integral for a given region, identify the bounds (limits) and the function(s) you need.
• For areas, typically, we integrate the difference between two functions across a specific interval.
• For specific calculations such as the center of mass, additional integrals set up using weighted functions are employed.

In our example, integrating the difference between the curves defined by functions like \( y = x^2 \) and \( y = x^4 \) finds the area of the region between these curves.

Understanding Calculus

Calculus is the mathematical study of changes, focusing on derivatives and integrals. It provides tools to analyze how things change and to solve problems involving motion, growth, and geometry, among others.

When determining the center of mass, calculus enables us to precisely calculate average positions by balancing these weighted sums.

This involves:

• Defining the functions and limits relevant to the problem space.
• Using derivatives to understand rates of change, though primarily we use integrals for this exercise.
• Applying formulas to find specific properties like centroids or centers of mass.

In this exercise, calculus is used beyond simple integration to perform these calculations, allowing us to determine detailed characteristics like the center of mass.

The Role of Symmetry

Symmetry is a powerful concept in geometry and calculus, simplifying calculations and improving understanding of mathematical and physical systems. In the context of finding the center of mass, symmetry allows us to quickly infer certain properties of the distribution of mass or area.

By using symmetry:

• We can often determine one of the coordinates of the center of mass without completing additional integrals.
• We can predict certain outcomes, such as balanced points or axes, reducing computation.
• It provides a sanity check for your calculations. If results don't align with expected symmetry, there might be an error.

In the initial problem, symmetry of the region bounded by \( y = x^2 \) and \( y = x^4 \) can help anticipate the center of mass location, as this region naturally has symmetry about certain axes, simplifying the identification of \( \overline{x} \) or \( \overline{y} \).