Problem 11 - Calculus for Biology and Medicine

Question

Use the trapezoidal rule to approximate each integral with the specified value of \(n\). $$ \int_{0}^{1} \exp (\sqrt{x}) d x, n=3 $$

Step-by-Step Solution

Answer

The approximate value of the integral using the trapezoidal rule with \( n = 3 \) is 1.6923.

Step 1: Understanding the Trapezoidal Rule

The trapezoidal rule is a numerical technique to approximate the definite integral of a function. If we have an integral \( \int_{a}^{b} f(x) \, dx \), and we want to approximate it using \( n \) subintervals, the trapezoidal rule formula is: \[ \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{2n} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right], \] where \( x_i = a + i \frac{b-a}{n} \).

Step 2: Define the parameters

For this problem, the integral \( \int_{0}^{1} \exp (\sqrt{x}) \, dx \) is given with \( n = 3 \). This means we are dividing the interval \([0, 1]\) into 3 equal parts. Hence, \( \Delta x = \frac{b-a}{n} = \frac{1}{3} \). The function is \( f(x) = \exp(\sqrt{x}) \).

Step 3: Calculate the x values

Since we have \( n = 3 \), we need to calculate \( x_0, x_1, x_2, \) and \( x_3 \). These values are given by \( x_i = i \cdot \frac{1}{3} \):- \( x_0 = 0 \)- \( x_1 = \frac{1}{3} \)- \( x_2 = \frac{2}{3} \)- \( x_3 = 1 \)

Step 4: Evaluate the function at each x value

Now we evaluate \( f(x) = \exp(\sqrt{x}) \) for each \( x_i \):- \( f(x_0) = \exp(\sqrt{0}) = \exp(0) = 1 \)- \( f(x_1) = \exp\left(\sqrt{\frac{1}{3}}\right) \approx 1.3956 \)- \( f(x_2) = \exp\left(\sqrt{\frac{2}{3}}\right) \approx 1.8221 \)- \( f(x_3) = \exp(\sqrt{1}) = \exp(1) \approx 2.7183 \)

Step 5: Apply the Trapezoidal Rule formula

Plug the values into the trapezoidal rule formula: \[ \int_{0}^{1} \exp(\sqrt{x}) \, dx \approx \frac{1}{2 \times 3} \left[ 1 + 2(1.3956 + 1.8221) + 2.7183 \right]. \]

Step 6: Simplify the Expression

Calculate the inner sum: - \( 2(1.3956 + 1.8221) = 6.4354 \)Now substitute back: \[ \int_{0}^{1} \exp(\sqrt{x}) \, dx \approx \frac{1}{6} \left[ 1 + 6.4354 + 2.7183 \right] = \frac{1}{6} \times 10.1537 \approx 1.6923. \]

Key Concepts

Numerical Integration

Numerical integration is a method used to estimate the value of a definite integral when an analytical solution is difficult or impossible to obtain. This approach divides the area under the curve into simple shapes that can be easily calculated. These shapes are usually rectangles, trapezoids, or other types of polygons, depending on the method used.

Some of the most popular numerical integration methods are

1. The Trapezoidal Rule
2. Simpson's Rule
3. Midpoint Rule

Each of these methods comes with its own formula and level of accuracy relative to the complexity of the function being integrated.

In essence, numerical integration offers an effective way to approximate integrals and quantify the area under curves. This method is widely used in fields like physics and engineering, where exact solutions might be impractical or unattainable.

Definite Integral

A definite integral represents the signed area under a curve between two limits on the x-axis. The notation for a definite integral of a function \( f(x) \) from \( a \) to \( b \) is \( \int_{a}^{b} f(x) \, dx \). It is a fundamental concept in calculus that allows for the calculation of areas, volumes, central points, and other critical quantities.

Unlike indefinite integrals, which represent a family of functions, definite integrals are numbers that describe specific cumulative totals or areas.

• The lower limit \( a \) and upper limit \( b \) define the interval over which the area is calculated.
• The process involves integrating the function across the interval, resulting in a concrete numerical value.

This concept is extremely important in applications across sciences and engineering, where it is used to determine quantities like mass, charge, and even rates of flow.

Composite Trapezoidal Rule

The Composite Trapezoidal Rule is an extension of the basic Trapezoidal Rule, used to approximate more complex integrals by dividing the integral interval into smaller subintervals. Each subinterval is treated with a trapezoid whose area can be easily calculated.

To apply this method:

• Divide the interval \([a, b]\) into \( n \) equal parts.
• Calculate \( x_i = a + i\frac{b-a}{n} \) for all subintervals.
• Compute \( f(x_i) \) for each subinterval point.
• Use the composite formula: \[ \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{2n} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \]

This technique provides the advantage of producing more accurate results than using a single interval. It's particularly useful when dealing with functions that are difficult to integrate analytically, as it breaks down the problem into manageable segments.