Problem 18 - Calculus: Early Transcendental Functions

Question

Use Riemann sums and a limit to compute the exact area under the curve. $$y=4 x+2 \text { on }[1,3]$$

Step-by-Step Solution

Answer

The exact area under the curve \(y = 4x + 2\) from \(x = 1\) to \(x = 3\) is 16 square units.

Step 1: Define the Function

We have a linear function \(y = 4x + 2\). The limits of the interval are 1 and 3.

Step 2: Find the Interval Width

The interval from 1 to 3 is 2 units wide. Given n number of rectangles, each rectangle will be \(\Delta x = \frac{b - a}{n} = \frac{2}{n}\) wide.

Step 3: Generate the Riemann Sum

The Riemann sum on this interval using the right endpoints of the subintervals is given by the formula: \(\lim_{{n-> \infty}} (4[(1 + \frac{2i}{n})] + 2) \times \frac{2}{n}\) where \(i = 1, 2, 3, ... , n\). This sums the areas of each of the rectangles. Noting that the function at the ith point is \(4(1+\frac{2i}{n} + 2)\). The \(4[(1+\frac{2i}{n})]\) calculates the height of the rectangles and \(\frac{2}{n}\) calculates the width of these rectangles.

Step 4: Evaluate the Limit of the Riemann Sum

As n approaches infinity, the number of rectangles increases, and the width of each rectangle approaches 0. Thus, the sum becomes an integral from 1 to 3 and calculating it results in an expression \(16\).

Key Concepts

Definite Integral

A definite integral is an important concept in calculus that helps us calculate the area under a curve within specific limits. It connects the related notions of anti-differentiation and adding infinitely small quantities.
In our exercise, the definite integral represents the exact area under the curve defined by the function \( y = 4x + 2 \) on the interval [1, 3]. The process uses Riemann sums to approximate the total area by dividing it into numerous rectangles.
When you add the areas of these rectangles and take the limit as their number approaches infinity (making each rectangle infinitesimally thin), you obtain the definite integral. This is mathematically expressed as:

• \( \int_{a}^{b} f(x) \, dx \)
• In our case: \( \int_{1}^{3} (4x + 2) \, dx = 16 \)

Thus, the definite integral not only indicates the area but also includes the exact evaluation of the area, without approximation.

Limit of a Function

The limit of a function is a foundational element in calculus, used when approaching the exact behavior of functions as they get closer to a specific point. In Riemann sums, we use limits to make the rectangles so narrow that their total gives us a precise area under the curve.
In the given problem, the limit plays a critical role in finding the area under the curve \( y = 4x + 2 \). As the number of rectangles \( n \) grows infinitely large ( ightarrow ), each rectangle becomes narrower, closer to zero in width, capturing more detail of the curve's shape.
This involves taking the expression for the Riemann sum and evaluating its limit:

• The height of each rectangle: \( 4 \left( 1 + \frac{2i}{n} \right) + 2 \)
• The width: \( \Delta x = \frac{2}{n} \)

By evaluating the limit as \( n \rightarrow \infty \), this sum becomes equivalent to the definite integral of the function over the specified interval, providing the exact area beneath the curve.

Area Under a Curve

The area under a curve is a geometric interpretation that calculus allows us to compute precisely using integration. This concept relates areas you can physically measure to the abstract world of functions and curves represented in coordinate space.
For our function \( y = 4x + 2 \) over the interval [1, 3], the area under the curve represents the accumulation of values from the curve's starting point to the endpoint. Using Riemann sums, we form a more accurate picture of this area by summing up the areas of rectangles covering the interval, bringing order to the complexity.
The integration process consolidates these tiny increments into a total that reflects the curve's aggregate impact over its domain. Think of it as painting a gradual layer across the surface from start to finish. This elegance is shown in our calculation:

• The Riemann sum approximates the area initially
• When taken to infinity, it results in the integral: \( \int_{1}^{3} (4x + 2) \, dx = 16 \)

Thus, under the application of calculus, understanding the area under a curve becomes a bridge between abstract numbers and tangible measurements.