Problem 35 - 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 \( n \to \infty \). $$ f(x) = \frac{1}{2}x + 4 $$ Interval \( [-1, 3] \)

Step-by-Step Solution

Answer

In the first part, use the formula \( \Delta x \cdot \sum_{i=1}^{n} f(x_i^{*}) \) where \( \Delta x = \frac{b - a}{n} \) and \(x_i^{*}\) is the x-coordinate of the ith rectangle in order to approximate the area under the curve. For the second part, use the definite integral \[ \int_{-1}^{3} \frac{1}{2}x + 4 \, dx \] to compute the exact area.

Step 1: setting up the formula

The formula to set-up the summation for the area of the rectangle is \[ \Delta x \cdot \sum_{i=1}^{n} f(x_i^{*}) \] where \( \Delta x = \frac{b - a}{n} \)], \(x_i^{*}\) is the x-coordinate of the ith rectangle. Since the number of rectangles is not given, we will look at the general form for the area using this approximation method.

Step 2: Plugging the function and the intervals

The intervals are given as [-1, 3] so \(a = -1\), \(b = 3\) and \(n\) needs to be calculated. We insert these values in the formula we got from Step 1. This is a general form to use when calculating.

Step 3: Computing the exact area using definite integral

The exact area under the curve will be computed by using definite integral. The limits of the integral will be \(a\) and \(b\), the bounds of the interval. With the values given, this will be \[ \int_{-1}^{3} \frac{1}{2}x + 4 \, dx \] By evaluating this integral, the exact area will be obtained.

Key Concepts

Rectangular Approximation Method

When we want to estimate the area under a curve, a common method is the Rectangular Approximation Method. This involves dividing the area into several rectangles and calculating the total area of these rectangles. The more rectangles we use, the closer our approximation will be to the actual area.

To do this, we need to determine the width of each rectangle, denoted as \( \Delta x \). This is calculated by dividing the length of the interval \([a, b]\) by the number of rectangles \(n\):

• \( \Delta x = \frac{b - a}{n} \)

Each rectangle's height is determined by the function \( f(x) \) evaluated at a specific point within that subinterval, often at the left endpoint, right endpoint, or midpoint.

Using this method, the sum of the areas of these rectangles gives us an approximation: \[ \sum_{i=1}^{n} f(x_i^{*}) \Delta x \] As \( n \) approaches infinity, this sum becomes a better estimate for the actual area under the curve.

Area Under a Curve

Finding the area under a curve is a fundamental concept in calculus. It can tell us important information about the behavior of functions in various fields like physics, economics, and biology. Using an integral is the most precise way to find this area.

Definite integrals help us compute this exact area. The integral \( \int_{a}^{b} f(x) \, dx \) calculates the area under the curve described by \( y = f(x) \) from \( x = a \) to \( x = b \). In the example provided, the function is \( f(x) = \frac{1}{2}x + 4 \), and we calculate:

• \( \int_{-1}^{3} \left(\frac{1}{2}x + 4\right) \, dx \)

Evaluating this integral provides the exact area under the curve for the specified interval. Calculating integrals allows for precision and represents the limit of our rectangular approximations as \( n \to \infty \).

Piecewise Function Learning

Piecewise functions are a great way to describe complex behaviors with simple rules. They define functions in pieces, using different expressions for different intervals. This is particularly useful when modeling real-world phenomena that change in character at certain points.

Though our example didn't involve a piecewise function directly, understanding them is crucial in learning how to apply integrals and approximation methods across different segments of a graph.

When working with piecewise functions, ensure each segment is integrated or approximated correctly. Each piece must be considered separately when calculating total area or using approximation methods, respecting the intervals they belong to.

• Recognize where function rules change.
• Integrate each part separately.
• Simplify to find the total area under the curve.

This approach ensures you handle each section of the function correctly, providing a complete understanding of the total behavior.