Problem 12 - Biocalculus Calculus for the Life Sciences

Question

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. $$\int_{1}^{5} x^{2} e^{-x} d x, \quad n=4$$

Step-by-Step Solution

Answer

The approximate value of the integral is 1.6175.

Step 1: Determine the Interval Width

First, we need to find the width of each subinterval, denoted as \( \Delta x \). Since the interval \([1, 5]\) has a length of \(5 - 1 = 4\), and we are dividing it into \(n = 4\) subintervals, each subinterval will have a width of \( \Delta x = \frac{4}{4} = 1\).

Step 2: Find the Midpoints

For the Midpoint Rule, we find the midpoint of each subinterval. With \( \Delta x = 1\), we set our subintervals as \([1, 2], [2, 3], [3, 4], [4, 5]\). The midpoints are:- For \([1, 2]: x_1 = 1.5\)- For \([2, 3]: x_2 = 2.5\)- For \([3, 4]: x_3 = 3.5\)- For \([4, 5]: x_4 = 4.5\)

Step 3: Evaluate the Function at Each Midpoint

We evaluate the function \( f(x) = x^2 e^{-x} \) at each midpoint:- \( f(1.5) = 1.5^2 e^{-1.5} \)- \( f(2.5) = 2.5^2 e^{-2.5} \)- \( f(3.5) = 3.5^2 e^{-3.5} \)- \( f(4.5) = 4.5^2 e^{-4.5} \)

Step 4: Calculate the Function Values

Calculating the function values:- \( f(1.5) \approx 0.7595 \)- \( f(2.5) \approx 0.5146 \)- \( f(3.5) \approx 0.2505 \)- \( f(4.5) \approx 0.0934 \)

Step 5: Apply the Midpoint Rule Formula

The Midpoint Rule formula is given by:\[ \int_{a}^{b} f(x) \, dx \approx \Delta x \sum_{i=1}^{n} f(x_i) \]Substituting the values:\[ \int_{1}^{5} x^2 e^{-x} \, dx \approx 1 \times (0.7595 + 0.5146 + 0.2505 + 0.0934) \]

Step 6: Compute the Approximation

Add the function values:\[ 0.7595 + 0.5146 + 0.2505 + 0.0934 = 1.6175 \]Finally, apply \( \Delta x = 1 \):\[ 1 \times 1.6175 = 1.6175 \]

Key Concepts

Midpoint Rule

The Midpoint Rule is a method used in numerical integration to approximate the value of an integral. It is especially useful when dealing with functions that are difficult or impossible to integrate analytically. By using the Midpoint Rule, we can estimate the area under a curve by dividing it into smaller, more manageable segments called subintervals. Each subinterval is then approximated by using a rectangle, where the height of the rectangle is determined by the function value at the midpoint of the interval.

The formula for the Midpoint Rule can be expressed as follows:

• Approximation = \(\Delta x \sum_{i=1}^{n} f(x_i) \)

where \(\Delta x\) is the width of each subinterval, \(n\) is the number of subintervals, and \(f(x_i)\) is the function value at the midpoint of the \(i\)-th subinterval. This method provides a better estimate compared to simple rectangle methods because it accounts for the function's behavior more accurately within each subinterval.

Integral Approximation

Integral approximation is the process of finding a numerical value close to the actual integral of a function. Sometimes, we come across integrals that do not have an exact answer due to the complexity of the function. In such cases, numerical methods, like the Midpoint Rule, come in handy.

Numerical integration methods allow us to estimate the area under the curve by considering discrete points and summing up individual contributions. The goal is to minimize the difference between the approximate and actual values.

When properly implemented, numerical methods can provide highly accurate approximations even for challenging functions, making them essential tools in scientific computation and engineering.

Subintervals and Midpoints

Subintervals and midpoints are fundamental to the Midpoint Rule as they determine the points at which function values will be estimated. A given integral with limits \([a, b]\) is divided into \(n\) equal parts, called subintervals, each having a width \(\Delta x = (b-a)/n\).

For each subinterval \([x_{i-1}, x_{i}]\), the midpoint is calculated using the formula:

• Midpoint of \(i\)-th subinterval: \(x_i = \frac{x_{i-1} + x_{i}}{2}\)

These midpoints \(x_i\) serve as the x-coordinates where the function is sampled. This approach tends to offer a better representation of the function's behavior within each subinterval compared to endpoints, thus achieving more precise integral approximations.

Exponential Decay Functions

Exponential decay functions, such as \(f(x) = x^2e^{-x}\), often arise in various scientific contexts. These functions demonstrate a rapid decrease in value as \(x\) increases, characteristic of processes like radioactive decay or cooling phenomena.

The general form of an exponential decay function is \(ae^{-bx}\), where \(a\) and \(b\) are constants. In the context of integration, these functions can present challenges due to their rapidly changing nature.

Using numerical integration methods like the Midpoint Rule can effectively help approximate their integrals, allowing calculations even when a symbolic solution is not easily obtainable. The function \(x^2e^{-x}\) combines polynomial and exponential components, which further complicates traditional integration, rendering numerical approaches crucial for estimation.