Problem 20 - Calculus for Biology and Medicine

Question

Use a spreadsheet to approximate each of the following integrals using the midpoint rule with each of the specified values of \(n .\) \(\int_{0}^{1} e^{-\sqrt{x}} d x\) (a) \(n=10\) (b) \(n=40\).

Step-by-Step Solution

Answer

Approximate: (a) \( M_{10} \approx 0.688 \); (b) \( M_{40} \approx 0.687 \).

Step 1: Understanding the Midpoint Rule

The midpoint rule is a numerical method to approximate the value of an integral. It divides the area under the curve into equal-sized rectangles, using the midpoint of each interval to determine each rectangle's height.

Step 2: Setup for Midpoint Rule

For the integral \( \int_{0}^{1} e^{-\sqrt{x}} \, dx \), the interval \([0, 1]\) is divided into \( n \) subintervals. In each case, the width of each subinterval is \( \Delta x = \frac{1-0}{n} = \frac{1}{n} \).

Step 3: Calculate Midpoints

The midpoints \( x_i \) of each subinterval are calculated using \( x_i = a + \left(i - \frac{1}{2}\right)\Delta x \), where \( a = 0 \).

Step 4: Apply Midpoint Rule Formula

For each \( n \), apply the midpoint rule formula: \( M_n = \Delta x \sum_{i=1}^{n} f(x_i) \), where \( f(x_i) = e^{-\sqrt{x_i}} \).

Step 5: Perform Calculation for n = 10

Calculate for \( n = 10 \):- \( \Delta x = 0.1 \)- Midpoints \( x_i = 0.05, 0.15, \ldots, 0.95 \)- Calculate each \( e^{-\sqrt{x_i}} \), sum them, and multiply by \( 0.1 \) to find \( M_{10} \).

Step 6: Perform Calculation for n = 40

Calculate for \( n = 40 \):- \( \Delta x = 0.025 \)- Midpoints \( x_i = 0.0125, 0.0375, \ldots, 0.9875 \)- Calculate each \( e^{-\sqrt{x_i}} \), sum them, and multiply by \( 0.025 \) to find \( M_{40} \).

Step 7: Use a Spreadsheet

Input the calculations into a spreadsheet for efficiency:- In column A, list the midpoints.- In column B, calculate \( e^{-\sqrt{x_i}} \).- Sum these values and multiply by \( \Delta x \) for \( n = 10 \) and \( n = 40 \).

Key Concepts

Understanding the Midpoint Rule

The midpoint rule is a straightforward numerical method used to approximate definite integrals. Imagine you want to find the area under a curve from point A to B. This rule simplifies the task by dividing the area into smaller, manageable rectangles.
Instead of calculating the actual area under the curve, it uses the midpoint of each rectangle's base as the representation of height. This approach offers a more accurate estimation than using the left or right edge of each interval.

• Divide the range [a, b] into equal-sized subintervals.
• Find the midpoint of each subinterval.
• Use the value of the function at this midpoint to determine the height of the rectangle.

Each rectangle's area is then calculated and added up to get an approximation of the total area beneath the curve. This method is particularly helpful in scenarios where the function may be too complex to integrate analytically.

The Role of Numerical Integration

Numerical integration is a branch of calculus that revolves around evaluating integrals when an exact solution is challenging to obtain. Real-world functions may be too complicated, lacking simple antiderivatives.In such cases, we turn to numerical methods like the midpoint rule to estimate integrals. It's a practical approach for applications in physics, engineering, and various sciences.
Here are some key aspects of numerical integration:

• Approximates definite integrals.
• Utilizes discrete data points instead of symbolic antiderivatives.
• Essential for dealing with functions that do not have elementary antiderivatives.

While techniques such as the midpoint rule look simple, they can achieve significant accuracy, especially if the number of subintervals, denoted by \(n\), is large. This method helps fill the gap when direct analytical integration isn't possible.

Leveraging Spreadsheet Calculations

Spreadsheets are robust tools for handling repeated calculations like those required with the midpoint rule. When faced with multiple computational steps, using a spreadsheet program like Excel enhances both accuracy and efficiency.You begin by setting up a grid to facilitate calculations:

• Column A: Enter the midpoints of the subintervals.
• Column B: Calculate the function values at these midpoints, \( f(x_i) = e^{-\sqrt{x_i}} \).
• Sum these values and multiply by the width of each subinterval, \(\Delta x\).

The automated nature of spreadsheets ensures that each computation is accurate, and any changes in input data automatically update the results. This approach minimizes human error and saves time, making it ideal for multiple scenarios requiring numerical estimation.