Problem 57 - Calculus Early Transcendentals

Question

Find the area of the following regions, expressing your results in terms of the positive integer \(n \geq 2\) The region bounded by \(f(x)=x\) and \(g(x)=x^{1 / n}\) for \(x \geq 0\)

Step-by-Step Solution

Answer

Question: Determine the area of the region bounded by the functions \(f(x) = x\) and \(g(x) = x^{\frac{1}{n}}\), for \(x \geq 0\) and positive integer \(n \geq 2\). Answer: The area of the region bounded by the functions is \(A = \frac{1}{2}- \frac{n}{n+1}\).

Step 1: Find the points of intersection of the two functions

We need to find the values of \(x\) where \(f(x) = g(x)\). So, we will set up an equation: $$ x = x^{\frac{1}{n}} $$ To solve this, we can take both sides to the power of \(n\): $$ x^n = x $$ This equation has two solutions for \(x\): \(x = 0\) and \(x = 1\). These are the points of intersection.

Step 2: Set up the definite integral

To find the area between the two functions, we'll integrate the difference of the functions within the interval of intersection, which is \([0, 1]\). The definite integral is: $$ A = \int_{0}^{1} (f(x) - g(x)) \, dx $$ Substituting the functions \(f(x) = x\) and \(g(x) = x^{\frac{1}{n}}\) : $$ A = \int_{0}^{1} (x - x^{\frac{1}{n}}) \, dx $$

Step 3: Perform the integration

Now we'll integrate the expression with respect to \(x\) using the power rule, which says that the integral of \(x^p\) is \(\frac{x^{p+1}}{p+1}\): $$ A = \left[\frac{x^2}{2} - \frac{x^{\frac{1}{n} + 1}}{\frac{1}{n} +1}\right]_{0}^{1} $$ Now, we will substitute the limits of integration: $$ A = \left[\frac{1^2}{2} - \frac{1^{\frac{1}{n} + 1}}{\frac{1}{n} +1}\right] - \left[\frac{0^2}{2} - \frac{0^{\frac{1}{n} + 1}}{\frac{1}{n} +1}\right] $$ Evaluating the expression, we get: $$ A =\left[\frac{1}{2} - \frac{1}{\frac{1}{n} +1}\right] - [0] = \frac{1}{2} - \frac{1}{\frac{1}{n} +1} $$

Step 4: Simplify the result

Finally, we will simplify the expression for the area: $$ A = \frac{1}{2} - \frac{1}{\frac{n+1}{n}} = \frac{1}{2}- \frac{n}{n+1} $$ Therefore, the area of the region bounded by the functions \(f(x) = x\) and \(g(x) = x^{\frac{1}{n}}\), for \(x \geq 0\) and positive integer \(n \geq 2\) is: $$ A = \frac{1}{2}- \frac{n}{n+1} $$

Key Concepts

Power Rule in Integration

When dealing with definite integrals, one of the most essential tools at your disposal is the Power Rule for Integration. This rule greatly simplifies the integration process for polynomial functions. The Power Rule states that the integral of a function of the form

• \( x^p \)

can be found using the formula:

• \[ \int x^p \, dx = \frac{x^{p+1}}{p+1} + C \]

where \( p eq -1 \) and \( C \) is the constant of integration. When applied to definite integrals, the constant is not necessary as we evaluate between two bounds.

In the context of our exercise, we evaluate the integral of \( x \) and \( x^{\frac{1}{n}} \) over the interval \([0, 1]\). Applying the Power Rule, the integral of \( x \) gives \( \frac{x^2}{2} \), while for \( x^{\frac{1}{n}} \), it yields \( \frac{x^{\frac{1}{n} + 1}}{\frac{1}{n} + 1} \). Hence, integrating these functions allows us to find the area between the curves, which is crucial for solving problems involving definite integration of power functions.

Area Between Curves

Calculating the area between two curves involves integrating the difference of the two functions over a certain interval. In many applications, this process involves several steps and helps us determine how much space lies between two given graphs on a coordinate plane.

To find the area between the given functions in our exercise, we use:

• \[ A = \int_{a}^{b} (f(x) - g(x)) \, dx \]

where:

• \( f(x) = x \)
• \( g(x) = x^{\frac{1}{n}} \)

In our case, the area is found over the interval \([0, 1]\), which are the points of intersection. By integrating the difference, \( x - x^{\frac{1}{n}} \), we can effectively measure the area enclosed between these two curves. This calculation results in the definite integral, indicating a fixed area value.

Intersection Points of Functions

Finding intersection points is a critical step in solving problems involving the area between curves. These points tell us where the curves meet, helping in determining the boundaries for integration.

In this exercise, we solve for intersection points by setting the functions equal:

• \( x = x^{\frac{1}{n}} \)

By raising both sides to the power of \( n \), we simplify the equation to \( x^n = x \). Solving this, we determine two intersection points:

• \( x = 0 \)
• \( x = 1 \)

These solutions provide the interval \([0, 1]\) for our definite integral, setting the boundaries essential for calculating the area between \( f(x) \) and \( g(x) \). Understanding how to derive these intersection points is fundamental when working with integration in calculus, especially for assigning proper limits to your integral.