Problem 30 - Calculus Volume 1

Question

In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? \([\mathrm{T}] L_{50}\) and \(R_{50}\) for \(y=\frac{x+1}{x^{2}-1}\) on the interval \([2,4]\)

Step-by-Step Solution

Answer

The area under the curve is between the left sum \( L_{50} \) and the right sum \( R_{50} \).

Step 1: Understand the Function and Interval

We need to graph and evaluate the function \( y = \frac{x+1}{x^2-1} \) over the interval \([2, 4]\). This includes observing any important features such as intercepts and asymptotes which can help in visualizing the curve.

Step 2: Compute the Grid Points and Subinterval Width

For 50 intervals from \(x=2\) to \(x=4\), the width of each subinterval is given by \( \Delta x = \frac{4-2}{50} = 0.04 \).

Step 3: Calculate the Left Riemann Sum \( L_{50} \)

Calculate the left endpoint sum: \( L_{50} = \sum_{i=0}^{49} \left( \frac{x_i + 1}{x_i^2-1} \right) \Delta x \), where \( x_i = 2 + i \times 0.04 \). Evaluate the function at each left endpoint and multiply by 0.04, then sum all the products.

Step 4: Calculate the Right Riemann Sum \( R_{50} \)

Calculate the right endpoint sum: \( R_{50} = \sum_{i=1}^{50} \left( \frac{x_i + 1}{x_i^2-1} \right) \Delta x \), where \( x_i = 2 + i \times 0.04 \). Evaluate the function at each right endpoint and multiply by 0.04, then sum all the products.

Step 5: Compare \( L_{50} \) and \( R_{50} \)

Evaluate the sums obtained from the left and right endpoint calculations to get \( L_{50} \) and \( R_{50} \). The area under the curve on the interval \([2, 4]\) is approximately between \( L_{50} \) and \( R_{50} \).

Key Concepts

Definite Integrals

A definite integral is a fundamental concept in calculus that helps to calculate the area under a curve within a specific interval. It is represented as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration. This notation tells us to find the area under the function \( f(x) \) from \( x = a \) to \( x = b \).
In practice, definite integrals allow us to compute total quantities, such as areas and volumes, from rate functions. They are closely related to the concept of antiderivatives. In the exercise, the area of interest under the curve \( y = \frac{x+1}{x^2-1} \) from 2 to 4 can be approximated using Riemann sums. These sums approach the value of the definite integral as we increase the number of intervals.

Subinterval Width

The subinterval width, often denoted as \( \Delta x \), is key when working with Riemann sums. It represents the width of each small subinterval into which the main interval is divided. Calculating \( \Delta x \) involves subtracting the starting x-value from the ending x-value and dividing by the desired number of subintervals.
In our example, with an interval from 2 to 4 split into 50 subintervals, the subinterval width is \( \frac{4-2}{50} = 0.04 \).
This small width gives a more accurate approximation of the area under the curve. Smaller subinterval widths generally lead to more precise results because they capture more detail in the curve's behavior over the interval.

Graphing Functions

Graphing a function is a valuable skill in understanding its behavior over a specific interval. A graph visually represents features like intercepts, maxima, minima, and asymptotes. These features can significantly influence the interpretation of a function's integral.
For the function \( y = \frac{x+1}{x^2-1} \), graphing shows us the curve's general shape and complexity. It may reveal vertical asymptotes, which are values of x where the function does not exist, such as points where the denominator equals zero. In practice, before using numerical approximations or calculus methods, graphing gives insight into important areas under consideration.

Numerical Integration

Numerical integration involves approximating the value of a definite integral when an exact form is difficult to obtain. One of the simplest methods is the Riemann sum, which uses shapes like rectangles to estimate the area under a curve.
Two common Riemann sum techniques include left endpoint sums and right endpoint sums. The left sum evaluates the function at the beginning of the subinterval, while the right sum evaluates at the end. Comparing these two gives an estimate of the integral's value.
In our exercise, 50 intervals are used, refined by computing both left \( L_{50} \) and right \( R_{50} \) endpoint sums. This method is efficient for irregular functions where algebraic methods might be less convenient.