Problem 56 - 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 \(g(x)=x^{n}\) for \(x \geq 0\)

Step-by-Step Solution

Answer

Answer: The area of the region bounded by the functions is given by \(A = \frac{1}{2} - \frac{1}{n+1}\).

Step 1: Find the Intersection Point(s)

Since we are given the functions \(f(x) = x\) and \(g(x) = x^n\), we need to find the x-coordinates where these functions intersect. This can be done by setting the two functions equal to each other and solving for \(x\), as follows: \(f(x) = g(x)\) \(x = x^n\) Now we can check the two cases when \(x=0\) and \(x=1\): - For \(x=0\): \(0 = 0^n\) - For \(x=1\): \(1 = 1^n\) These are true for any positive integer \(n \geq 2\). So, the functions intersect at the points \((0,0)\) and \((1,1)\) and these points have \(x\)-coordinates of 0 and 1, respectively.

Step 2: Set up and Evaluate the Integral

Now that we have the intersection points, we can set up the integral representing the area between the two curves. For this problem, the area is equal to: \(A = \int_{0}^{1}(f(x) - g(x))\,dx\) We can now substitute the given functions into the integral: \(A = \int_{0}^{1} (x - x^n)\,dx\) In order to evaluate the integral, we will apply integral rules and find the antiderivative of each term. The antiderivative of \(x\) is \(\frac{1}{2}x^2\), and of \(x^n\) is \(\frac{1}{n+1}x^{n+1}\). Using the Fundamental Theorem of Calculus, we find the area between the two curves: \(A = \left[\frac{1}{2}x^2 - \frac{1}{n+1}x^{n+1}\right]_{0}^{1}\) Now, we will substitute the \(x\)-coordinates of the intersection points and subtract: \(A = \left(\frac{1}{2}(1)^2 - \frac{1}{n+1}(1)^{n+1}\right) - \left(\frac{1}{2}(0)^2 - \frac{1}{n+1}(0)^{n+1}\right)\) After simplifying the expression, the answer is: \(A = \frac{1}{2} - \frac{1}{n+1}\) This is the area of the region bounded by the functions \(f(x)=x\) and \(g(x)=x^n\) for \(x\geq 0\) and positive integer \(n\geq 2\).

Key Concepts

Integral Calculus

Integral calculus is a fundamental branch of mathematics that deals with the concept of integration, which is essentially finding the whole given its parts. When calculating the area between curves, as in the given problem, integral calculus provides the tools needed for computing these areas.
To find the area between two curves, we essentially take the integral of the difference of the two functions. This process helps in summing up infinitely small slices under the curve into a single value, representing the total area we are interested in.
In the problem, we are integrating the function derived from subtracting the given functions: \(A = \int_{0}^{1} (x - x^n)\,dx\). Here, integral calculus allows us to transform a geometric question into an algebraic formula that can be calculated concretely.

Intersection Points

Intersection points are where two or more graphs meet on a coordinate plane. Finding these points is crucial in determining the limits of integration when calculating the area between two curves.
In the exercise, the given functions are set equal to find these points: \(x = x^n\). Solving this equation gives potential solutions – notably at \(x=0\) and \(x=1\), since for any \(n \geq 2\), \(x^n\) will always satisfy this equality at these points.
These intersection points, \( (0,0) \) and \( (1,1) \), are the boundaries for the region where the curves enclose an area, which is why they are used as the limits of integration. They essentially mark the beginning and end of the region over which the area is calculated.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, showing that one can reverse the process of differentiation through integration, and vice versa.
This theorem is crucial for solving problems involving areas under a curve, like in the current exercise. It provides a powerful method to evaluate the definite integrals needed to find the area.
When we apply this to our integral \(\int_{0}^{1} (x - x^n)\, dx\), the theorem tells us that the value of this integral can be computed by finding the antiderivative of the integrand first, then calculating the difference between its values at the upper and lower limits of integration: \[A = \left[\frac{1}{2}x^2 - \frac{1}{n+1}x^{n+1}\right]_{0}^{1}\].
This is why once we find the antiderivative, we only need to substitute the intersection points \(x=0\) and \(x=1\) into the antiderivative followed by subtraction to find the area, showing the theorem's practical utility in solving this problem.