Problem 13 - Calculus for Biology and Medicine

Question

Approximate $$ \int_{-1}^{2} e^{-x} d x $$ using three equal subintervals.

Step-by-Step Solution

Answer

The integral is approximately \(2.7946\).

Step 1: Identify the Interval

The interval for integration is from \(-1\) to \(2\).

Step 2: Determine Subintervals

Divide the interval \([-1, 2]\) into three equal subintervals. Each subinterval will have a width \(h = \frac{2 - (-1)}{3} = 1\).

Step 3: Calculate the Endpoints

The endpoints are \(-1\), \(0\), \(1\), and \(2\).

Step 4: Approximate with Trapezoidal Rule

The Trapezoidal Rule formula is \(\approx \frac{h}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + f(x_3)]\). Calculate the values: \( f(x) = e^{-x} \) at points \(-1, 0, 1, 2\).

Step 5: Evaluate Exponential Values

Calculate \(f(-1) = e^{1}\), \(f(0) = e^{0} = 1\), \(f(1) = e^{-1}\), and \(f(2) = e^{-2}\).

Step 6: Apply the Trapezoidal Formula

Substitute these values into the Trapezoidal Rule: \(\approx \frac{1}{2} [e^{1} + 2 \cdot 1 + 2 \cdot e^{-1} + e^{-2}]\).

Step 7: Final Calculation

Perform the calculations: \(\approx \frac{1}{2} [2.71828 + 2 + 2(0.36788) + 0.13534]\). Sum them: \(\approx \frac{1}{2} [5.58918]\).

Step 8: Compute the Result

Divide by 2 to find the result: \( \approx 2.7946 \).

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a method used in calculus to approximate the definite integral of a function. It is particularly useful when you need a quick estimation without performing exact calculations. The process involves splitting the region under a curve into trapezoidal sections instead of using the typical rectangles seen in Riemann sums. This leads to a more accurate approximation, especially when dealing with curves rather than straight lines.

In practical terms:

• Divide the interval: Break down the entire range of integration into smaller subintervals of equal width.
• Calculate endpoints: Determine the exact values at these endpoints.
• Apply the formula: Use the Trapezoidal Rule formula \[\text{Approximation} = \frac{h}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + f(x_n)] \]
  where \(h\) is the width of each subinterval and \(x_0, x_1, \ldots, x_n\) are the endpoints of the subintervals.

The trapezoidal shape accounts for the slope between two points and helps in smoothing out errors that might occur with other approximation methods. This makes it an important tool in numerical calculus.

Exponential Function

Exponential functions are a type of mathematical function in which a constant base is raised to a variable exponent. The most common base for an exponential function is Euler's number, \(e\), approximately equal to 2.71828.

The exponential function \(f(x) = e^x\) is noteworthy due to its unique property: the rate of change (derivative) of \(e^x\) is equal to \(e^x\) itself. This property is crucial in many areas of calculus, especially in problems related to growth and decay, such as population growth, radioactive decay, and more.

When dealing with integrations involving exponential functions:

• Integration property: The integral of an exponential function \(\int e^{-x} \, dx\) is straightforward. It follows from basic integration rules.
• Extensive use in approximations: Exponential functions occur frequently, making it essential to understand their behavior and how they can be manipulated within numerical methods like the Trapezoidal Rule.

In the context of numerical approximation, assessing the values of the exponential function at key points can provide the data needed to apply these methods accurately.

Numerical Approximation

Numerical approximation is a broad field that focuses on finding approximate solutions to mathematical problems which might be difficult or even impossible to solve analytically. Integration is a classic example where numerical methods like the Trapezoidal Rule become invaluable.

The reason for numerical approximations:

• Simplicity: Some integrals are too complex to have a closed-form solution, making numerical methods a simple alternative.
• Flexibility: Numerical methods can handle functions that are discontinuous or do not integrate neatly using traditional calculus techniques.
• Practical necessity: In real-world applications, getting an exact answer is often less important than achieving an accurate approximation quickly and efficiently.

In using numerical approximations like the Trapezoidal Rule, we aim to balance error and efficiency. While the results are not exact, they often suffice for engineering, physics, and other applied sciences, where the exact integral is less crucial than understanding the trends or behaviors modeled by the integral.