Problem 12 - Calculus for Biology and Medicine

Question

Use the trapezoidal rule to approximate each integral with the specified value of \(n\). $$ \int_{0}^{1} \sin (\sqrt{x}) d x, n=4 $$

Step-by-Step Solution

Answer

The approximate value of the integral is 0.4900 using the trapezoidal rule with 4 sub-intervals.

Step 1: Understand the Concept

The trapezoidal rule is a numerical method to approximate the integral of a function. By dividing the interval into smaller sub-intervals, it approximates the area under the curve as a series of trapezoids. The more sub-intervals (7) we use, the more accurate our approximation becomes.

Step 2: Set Up the Parameters

We are asked to approximate the integral \( \int_{0}^{1} \sin (\sqrt{x}) \ dx \) using 4 sub-intervals (\( n = 4 \)). This means we divide the interval [0,1] into 4 equal parts. Since the total interval size is 1, each sub-interval (\( h \)) will have a length of \( h = \frac{1-0}{4} = 0.25 \).

Step 3: Calculate the Subinterval Endpoints

Determine the endpoints of each sub-interval. The endpoints for \( n=4 \) are: \( x_0 = 0 \), \( x_1 = 0.25 \), \( x_2 = 0.5 \), \( x_3 = 0.75 \), and \( x_4 = 1 \).

Step 4: Evaluate the Function at Each Endpoint

Compute the value of the function \( \sin(\sqrt{x}) \) at each of the endpoints. We find: - \( f(x_0) = \sin(\sqrt{0}) = 0 \)- \( f(x_1) = \sin(\sqrt{0.25}) \approx \sin(0.5) \approx 0.4794 \)- \( f(x_2) = \sin(\sqrt{0.5}) \approx \sin(0.7071) \approx 0.6494 \)- \( f(x_3) = \sin(\sqrt{0.75}) \approx \sin(0.866) \approx 0.761 \)- \( f(x_4) = \sin(\sqrt{1}) = \sin(1) \approx 0.8415 \)

Step 5: Apply the Trapezoidal Rule Formula

The trapezoidal rule formula for approximating the integral is: \[ T_n = \frac{h}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)] \]Substitute the function values and \( h \): \[ T_4 = \frac{0.25}{2} [0 + 2(0.4794) + 2(0.6494) + 2(0.761) + 0.8415] \]

Step 6: Perform the Calculations

Calculate inside the brackets:- Sum the products: \( 0 + 2(0.4794) + 2(0.6494) + 2(0.761) + 0.8415 = 3.9202 \)- Now compute the full expression: \( T_4 = \frac{0.25}{2} \cdot 3.9202 = 0.4900 \)

Key Concepts

Numerical Integration

Numerical integration is all about finding an approximate value of an integral that might be difficult to solve analytically. This technique provides a practical way to calculate the area under a curve or the accumulated quantity over an interval.
While the fundamental theorem of calculus connects the concept of an integral to antiderivatives, numerical methods are crucial when we either can't find an antiderivative or the function involves complex expressions. For instance, - Functions that involve irrational exponents, - Functions without easy-to-compute primitives, - Complex curves unsuitable for standard integration methods.
Here's where methods like the trapezoidal rule come into play, offering a straightforward technique by dividing the function into simpler shapes (trapezoids), allowing us to estimate the integral's value.

Approximate Integral

The concept of an approximate integral means finding a close estimate to the value of an integral, especially when an exact solution is impractical.
This is performed by breaking the integration process into manageable parts, each approximated by known geometric shapes. The trapezoidal rule, for instance, uses trapezoids because they offer a balance of simplicity and accuracy.
In the example of approximating \(\int_{0}^{1} \sin(\sqrt{x}) \ dx\) with \(n=4\):

• The integral is estimated by cutting the interval [0, 1] into four parts.
• Each part is transformed into a trapezoid where the top side follows the curve as closely as possible.
• The area of these trapezoids combined offers a good approximation of the total area beneath the curve.

The more subdivisions one uses, the closer the approximate integral gets to the exact value, demonstrating this method's utility in real-world calculations.

Sub-intervals

Sub-intervals are the smaller segments into which the entire range of integration is divided. They play a crucial role in numerical integration methods like the trapezoidal rule by defining the parts of the curve approximated by simple geometric figures.
In our exercise, the interval [0,1] is divided into 4 sub-intervals, each 0.25 units wide.
To break it down:

• Each sub-interval corresponds to a segment between two consecutive endpoints, such as \(x_0\) and \(x_1\).
• These endpoints are crucial because at each one, the function \(\sin(\sqrt{x})\) is evaluated to give the heights of the trapezoidal sides.
• Organizing the integration process into sub-intervals simplifies the calculation, turning the complex integral into a sequence of straightforward additions and multiplications.

The division into sub-intervals not only facilitates computation but also helps understand how the trapezoidal approximation manages to capture the area under the curve.