Problem 10 - Calculus for Biology and Medicine

Question

Use the trapezoidal rule to approximate each integral with the specified value of \(n\). $$ \int_{-1}^{0} \sin \left(x^{2}\right) d x, n=5 $$

Step-by-Step Solution

Answer

The integral is approximately 0.3139 using the trapezoidal rule with \( n = 5 \).

Step 1: Understand the Trapezoidal Rule

The trapezoidal rule approximates the integral of a function by dividing the area under the curve into trapezoids rather than rectangles. The formula for the trapezoidal rule with \( n \) subintervals is: \[ \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{2n} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right] \] where \( x_i \) are the endpoints of the subintervals.

Step 2: Determine Values for a, b, and n

For the integral \( \int_{-1}^{0} \sin(x^2) \, dx \), the values are: \( a = -1 \), \( b = 0 \), and \( n = 5 \). These values will be used to apply the trapezoidal rule.

Step 3: Calculate the Width of Each Interval

The width of each subinterval \( h \) is given by \( h = \frac{b-a}{n} = \frac{0 - (-1)}{5} = \frac{1}{5} = 0.2 \).

Step 4: Identify the x-values for Trapezoidal Approximations

With \( h = 0.2 \), the x-values are: \( x_0 = -1 \), \( x_1 = -0.8 \), \( x_2 = -0.6 \), \( x_3 = -0.4 \), \( x_4 = -0.2 \), and \( x_5 = 0 \). These points are the endpoints of the subintervals.

Step 5: Compute Function Values at the x-values

Calculate \( f(x_i) = \sin(x_i^2) \) for each \( x_i \):- \( f(x_0) = \sin((-1)^2) = \sin(1) \)- \( f(x_1) = \sin((-0.8)^2) = \sin(0.64) \)- \( f(x_2) = \sin((-0.6)^2) = \sin(0.36) \)- \( f(x_3) = \sin((-0.4)^2) = \sin(0.16) \)- \( f(x_4) = \sin((-0.2)^2) = \sin(0.04) \)- \( f(x_5) = \sin((0)^2) = \sin(0) = 0 \)

Step 6: Apply the Trapezoidal Rule Formula

Use the trapezoidal rule formula to calculate the approximate integral:\[\int_{-1}^{0} \sin(x^2) \, dx \approx \frac{0-(-1)}{2 \times 5} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + f(x_5) \right]\]Which simplifies to:\( \frac{1}{10} \left[ \sin(1) + 2 \sin(0.64) + 2 \sin(0.36) + 2 \sin(0.16) + 2 \sin(0.04) + 0 \right] \).

Step 7: Evaluate the Sine Functions and Compute the Sum

Calculate the individual sine values:- \( \sin(1) \approx 0.8415 \)- \( \sin(0.64) \approx 0.5974 \)- \( \sin(0.36) \approx 0.3520 \)- \( \sin(0.16) \approx 0.1593 \)- \( \sin(0.04) \approx 0.0399 \)Substitute these values back: \[\frac{1}{10} \left[ 0.8415 + 2(0.5974) + 2(0.3520) + 2(0.1593) + 2(0.0399) \right]\].

Step 8: Complete the Final Calculation

Simplify and calculate the expression:\[ \frac{1}{10} \left[ 0.8415 + 1.1948 + 0.7040 + 0.3186 + 0.0798 \right] = \frac{1}{10} \times 3.1387 \approx 0.3139 \]

Step 9: Conclusion: Approximate Value of the Integral

Therefore, the approximate value of the integral \( \int_{-1}^{0} \sin(x^2) \, dx \) using the trapezoidal rule with \( n = 5 \) is approximately 0.3139.

Key Concepts

Numerical Integration

Numerical integration is a powerful tool used to approximate the value of definite integrals, especially when an antiderivative is difficult or impossible to find analytically. It comes in handy for complex functions or when exact solutions are not viable. The primary aim of numerical integration is to estimate the area under a curve over a specified interval. Several methods can be employed for numerical integration, including techniques like:

• Trapezoidal Rule
• Simpson's Rule
• Midpoint Rule

The choice of method often depends on the required accuracy and the nature of the function. Each method subdivides the interval into smaller parts, simplifies the curve over each subinterval, and sums the resulting areas to approximate the total. Numerical integration is indispensable in fields like physics and engineering, where exact integrals are often impractical.

Approximate Integral

An approximate integral is an estimate of the value of an integral, calculated using numerical methods rather than analytical techniques. Approximations are essential when dealing with functions that are complex or do not have a simple antiderivative. The Trapezoidal Rule is one such technique to approximate integrals efficiently.The trapezoidal rule works by:

• Dividing the interval into multiple smaller subintervals
• Approximating the area under the curve with trapezoids
• Adding the areas of these trapezoids to find an estimate of the total integral

For example, the exercise used the trapezoidal rule with the function \(\sin(x^2)\) from \\(-1\) to \\(0\). By calculating the area of each trapezoid, the method provides a reliable estimate of the integral. The method reduces error by increasing the number or size of trapezoids, ultimately enhancing precision while keeping calculations simple.

Calculus for Biology

Calculus finds numerous applications in biology, aiding in the understanding of various natural processes. Numerical methods, like the Trapezoidal Rule, allow biologists to analyze phenomena that involve integration when precise formulas are difficult to obtain. Examples of applications include:

• Modeling population dynamics
• Calculating the distribution of resources among organisms
• Understanding biological growth rates

Calculus helps in modeling the way biological substances change over time or space. For instance, the rate of an enzyme reaction or the spread of a population can be studied using integrals. Additionally, numerical integration becomes particularly beneficial in biological research when dealing with experimental data that yield functions hard to integrate exactly. This way, biologists obtain practical insights without delving into complex calculus, bridging the gap between theoretical mathematics and real-world biological concerns.