Problem 50 - Calculus for Biology and Medicine

Question

Express the definite integrals as limits of Riemann sums. $$ \int_{1}^{3} e^{-2 x} d x $$

Step-by-Step Solution

Answer

Express \( \int_{1}^{3} e^{-2x} \, dx \) as \( \lim_{n \to \infty} \sum_{i=1}^{n} e^{-2 - \frac{4i}{n}} \frac{2}{n} \).

Step 1: Understanding the Integral

Our task is to express the definite integral \( \int_{1}^{3} e^{-2x} \, dx \) as a limit of Riemann sums. This involves partitioning the interval and approximating the area under the curve \( e^{-2x} \) using rectangles.

Step 2: Setting Up the Partition

We partition the interval \([1, 3]\) into \(n\) subintervals of equal width, \(\Delta x = \frac{3-1}{n} = \frac{2}{n}\). Each point in the partition is given by \(x_i = 1 + i\Delta x = 1 + i\frac{2}{n}\), where \(i\) ranges from 0 to \(n\).

Step 3: Defining the Riemann Sum

The Riemann sum for the function \(f(x) = e^{-2x}\) over the interval \([1, 3]\) using right endpoints is given by \( \sum_{i=1}^{n} e^{-2(1 + i\frac{2}{n})} \Delta x = \sum_{i=1}^{n} e^{-2 - \frac{4i}{n}} \frac{2}{n} \).

Step 4: Taking the Limit

The definite integral can be expressed as a limit of the Riemann sum as the number of subintervals approaches infinity: \( \int_{1}^{3} e^{-2x} \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} e^{-2 - \frac{4i}{n}} \frac{2}{n} \).

Step 5: Conclusion

Thus, the definite integral is represented as a limit of Riemann sums: \( \lim_{n \to \infty} \sum_{i=1}^{n} e^{-2 - \frac{4i}{n}} \frac{2}{n} \).

Key Concepts

Definite Integrals

A definite integral is a fundamental concept in calculus used to calculate the net area under a curve, often representing the total accumulation of a quantity. Specifically, the definite integral \( \int_{a}^{b} f(x) \, dx \) calculates the area under the function \( f(x) \) from \( x = a \) to \( x = b \). This area is bounded by the x-axis and the graph of the function. The result of a definite integral is a number that can represent, for example, distance, area, or accumulated change.

• Boundaries: \( a \) and \( b \) are the limits of integration and define the interval over which the area is calculated.
• Function: \( f(x) \) is the integrand or the curve being analyzed.

In practical terms, definite integrals are widely used in physics, engineering, and economics to model real-world situations.

Partition of Interval

Dividing an interval into subintervals is an essential step for computing a Riemann sum, leading toward evaluating a definite integral. A partition of an interval \([a, b]\) is a division of this interval into smaller subintervals, typically of equal length. For ease of computation, these segments help define points where approximations (like rectangles or trapezoids) are drawn beneath or over the curve of the function.

• Segmentation: If the interval \([a, b]\) is divided into \( n \) small parts, each of these parts is denoted as \( \Delta x \), which stands for the width of a subinterval.
• Calculation of \( \Delta x \): This is calculated as \( \Delta x = \frac{b-a}{n} \).
• Subinterval Points: Points dividing the subintervals are chosen, often as \( x_i = a + i \cdot \Delta x \).

For each subinterval, these points become critical references to evaluate the function's value which influences the calculation of the rectangular approximation or other, more complex, approaches.

Limit of Riemann Sum

A cornerstone of definite integrals is the limit of a Riemann sum. This concept considers how the sum of areas of shapes, like rectangles, converge to the exact area under a curve as the number of shapes becomes infinitely large and their width infinitesimally small.

• Riemann Sum Formula: A Riemann sum \( \sum_{i=1}^{n} f(x_i^*) \Delta x \) gives the approximate area under a function. Here, \( x_i^* \) are the sample points.
• Taking the Limit: As \( n \to \infty \), this sum approaches the true value of the definite integral: \( \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x = \int_{a}^{b} f(x) \, dx \).

The process of taking this limit essentially tightens the approximation until it accurately represents the exact area. It highlights the power of infinitesimal calculus to precisely measure quantities that are otherwise complex to calculate directly.

Rectangular Approximation

Rectangular approximation is a technique to estimate the area under a curve using rectangles. This method plays a crucial role in understanding the mechanics of an integral by breaking down the area into manageable units. There are different approaches to setting up these rectangles, most notably using left endpoints, right endpoints, or midpoints of subintervals.

• Right Endpoint Method: The right endpoint of each subinterval in a partition determines the height of the rectangle, often resulting in an overestimation or underestimation of the actual area.
• Calculation: For a function \( f(x) \) over an interval \([a, b]\), using right endpoints, the Riemann sum can be expressed as \( \sum_{i=1}^{n} f(x_i) \Delta x \) where \( x_i \) represents the right endpoint of each subinterval.
• Flexibility: Depending on the approach (left, right, or midpoint), the approximation's accuracy can vary, providing insights or leading to exploration of other integral approximation methods.

Despite being an approximation, this method is foundational in calculus education, providing the intuition necessary for understanding how integrals are calculated and applied across various fields.