Problem 34 - Precalculus with Limits

Question

In Exercises 29-36, complete the table using the function \( f(x) \), over the specified interval [a, b], to approximate the area of the region bounded by the graph of the \( y = f(x) \), the x-axis, and the vertical lines \(x=a\) and \(x=b\) and using the indicated number of rectangles. Then find the exact area as \( n \to \infty \). $$ f(x) = x^2 + 1 $$ Interval \( [4, 6] \)

Step-by-Step Solution

Answer

The exact area under the curve \( f(x) = x^2 + 1 \) from \( x=4 \) to \( x = 6 \) is found using the integral \( \int_{4}^{6} f(x) dx \).

Step 1: Calculate Δx

The width, or \( \Delta x \), of each rectangle is calculated by subtracting the lower limit of the interval from the upper limit and dividing by \( n \). Therefore, \( \Delta x = (6 - 4)/n = 2/n \).

Step 2: Calculate the Total Area of Rectangles

The total area of all the rectangles within the given interval can be calculated by adding up the individual areas of each rectangle. This is accomplished by evaluating 'f' at each x-value and multiplying by the width, \( \Delta x \). This gives the formula \( A \approx \sum_{i=1}^{n} f(x_{i}) \Delta x \) where \( x_{i} = 4 + i \Delta x \), for \( i \in [0, n] \).

Step 3: Approximate as n goes to infinity

As \( n \rightarrow \infty \), the sum transforms into an integral, and \( \Delta x \) becomes \( dx \). Therefore, the exact area under the curve is given by the integral \( \int_{4}^{6} f(x) dx\). Evaluating \( f(x) = x^2 + 1\) from 4 to 6, we find the exact area under the curve.

Key Concepts

Riemann Sums

Riemann Sums are a method for approximating the area under a curve. Imagine slicing the area under a curve into several vertical rectangles. The concept is simple: the more rectangles you use, the more accurate your approximation becomes. When creating a Riemann Sum, decide how many rectangles you want to use for your approximation. Each rectangle will have the same width, denoted as \( \Delta x \), and this width is determined by dividing the difference from the upper and lower limits of the interval by the number of rectangles.
To calculate the total area of these rectangles, evaluate the function at each specific point, usually the right, left, or midpoint of each rectangle, and multiply by the width. This gives the approximate area:

• \( A \approx \sum_{i=1}^{n} f(x_{i}) \Delta x \)
• Where \( x_{i} = a + i \Delta x \)
• The interval is \([a, b]\)

As you increase the number of rectangles \( n \), the approximation improves, and as \( n \to \infty \), the Riemann Sum approaches the definite integral of the function, offering an exact area.

Area Under a Curve

The area under a curve within a given interval is a fundamental concept in calculus. It represents the total accumulation of a quantity, and can be found using a definite integral. In practical applications, understanding how to find this area helps in quantifying various physical concepts like distance, work, and probability.
Imagine a curve defined by a function \( f(x) \). To find the area enclosed by this curve, the x-axis, and the vertical lines at \( x=a \) and \( x=b \), use the definite integral notation \( \int_{a}^{b} f(x) \, dx \). This process involves evaluating the function between these interval points:

• Integrating the area returns a result representing the exact sum of infinite tiny slices under the curve.
• This method provides greater precision than a Riemann Sum with a finite number of rectangles.

This integral evaluates to the net area between the function and the x-axis over the specified interval, taking into account parts of the curve that may lie below the x-axis, which get subtracted from the total.

Calculus

Calculus is a branch of mathematics that focuses on the study of rates of change and accumulation. It introduces concepts like limits, derivatives, and integrals to explore functions and their behaviors comprehensively. The two main branches of calculus are differential calculus and integral calculus:

- **Differential Calculus** investigates the rate of change or slope of curves through derivatives. It helps you find how things change quickly in a specific instance. - **Integral Calculus**, on the other hand, deals with accumulation or total values, such as areas under curves, through integrals. Calculus provides powerful tools for modeling and solving problems in sciences, engineering, and beyond. With these tools, you can handle complex systems and dynamic patterns.

• It transforms problems involving motion, change, and growth into a manageable mathematical framework.
• From measuring the changing speed of an object to computing the growth of investments, calculus offers a systematic approach.

By mastering calculus, you gain insight into a wide range of functions and learn to apply mathematical theory to real-life scenarios efficiently.