Problem 21 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Exercise 9.4 .21 Let \(x\) be a number in \(\left[0, \frac{\pi}{2}\right]\). Use the trigonometric identity $$\sum_{k=1}^{n} \cos (k \times \theta)=\frac{\sin \left(n \times \theta+\frac{\theta}{2}\right)-\sin \left(\frac{\theta}{2}\right)}{2 \sin \left(\frac{\theta}{2}\right)}$$ and $$\lim _{h \rightarrow 0} \frac{\sin (h)}{h}=1$$ to compute from Definition I the integral $$\int_{0}^{x} \cos (t) d t=\sin x$$

Step-by-Step Solution

Answer

The integral \( \int_0^x \cos(t) \, dt \) equals \( \sin(x) \).

Step 1: Understand the Integral

Our task is to compute the definite integral \( \int_0^x \cos(t) \ dt \), which should equal \( \sin(x) \) according to the fundamental theorem of calculus.

Step 2: Use Trigonometric Identity

We will use the given trigonometric identity to represent the sum of cosines. Note that for solving the integral itself, this sum is not directly involved, but understanding similar expressions helps understand the relationship between cosine and sine expressions.

Step 3: Recognize the Definition of the Integral

From Definition I, the integral \( \int_0^x \cos(t) \ dt \) is defined as the limit of Riemann sums as the partition becomes infinitely fine.

Step 4: Apply Limit Definition

Express the integral using the limit definition: \( \int_0^x \cos(t) \, dt = \lim_{n \to \infty} \sum_{k=1}^n \cos(t_k) \Delta t \), where \( \Delta t = \frac{x}{n} \) and \( t_k = \frac{kx}{n} \).

Step 5: Evaluate Riemann Sum

Compute the Riemann sum: \( \sum_{k=1}^{n} \cos\left( \frac{kx}{n} \right) \Delta t = \sum_{k=1}^n \cos\left( \frac{kx}{n} \right) \frac{x}{n} \).

Step 6: Use Continuous Limit

Recognize that as \( \Delta t = \frac{x}{n} \to 0 \), the sum converges to the integral. We use \( \lim_{h \to 0} \frac{\sin(h)}{h} = 1 \) to relate the integral of cosine to a sine function limit.

Step 7: Conclusion Using Limits

Because of the connection through derivative property of sine and cosine, the integral \( \int_0^x \cos(t) \ dt \) simplifies to \( \sin(x) - \sin(0) = \sin(x) \), proving the result.

Key Concepts

Trigonometric Identity

To solve the problem, it's crucial to understand the role trigonometric identities play in calculus. A trigonometric identity is an equality involving trigonometric functions that holds for any angle. In this exercise, you are given a specific identity: \( \sum_{k=1}^{n} \cos(k \times \theta)=\frac{\sin \left(n \times \theta+\frac{\theta}{2}\right)-\sin \left(\frac{\theta}{2}\right)}{2 \sin \left(\frac{\theta}{2}\right)} \).
This identity helps illustrate the relationships between different trigonometric functions. While it isn't directly used to solve the integral, recognizing such expressions enhances understanding of how \cos(x) and \sin(x) are related.

• Trigonometric identities are useful in simplifying complex expressions.
• They help in converting sums involving trigonometric functions into simpler forms.
• Understanding these identities can ease solving integrals involving trigonometric functions.

This particular identity leverages the sum of cosine functions and relates it to sine, laying foundational knowledge for various calculus applications.

Definite Integral

A definite integral is a fundamental concept in calculus representing the area under a curve. Here, we are tasked with evaluating the definite integral \( \int_0^x \cos(t) \, dt \), which interprets as the area under the \cos(t) graph from 0 to \(x\).
Through integration, we find that this area is precisely equal to \(\sin(x)\).

• Definite integrals have bounds (0 to \(x\) in this case) that define the interval over which you integrate.
• This integral results in an exact numerical value reflecting the accumulated quantity the function represents.
• Integrals can often be solved using antiderivatives, which are functions whose derivative is the integrand.

The core aim of solving a definite integral often involves applying properties like the Fundamental Theorem of Calculus to transition from the function to its accumulation—a critical skill in applying calculus.

Riemann Sums

Riemann sums are a method for approximating the area under a curve, representing the integral of a function. The sum functions as a pivotal step towards calculating definite integrals.
The Riemann sum for the integral \( \int_0^x \cos(t) \, dt \) is expressed as \( \lim_{n \to \infty} \sum_{k=1}^n \cos(t_k) \Delta t \). Here, \( \Delta t = \frac{x}{n} \) and \( t_k = \frac{kx}{n} \).

• In Riemann sums, the interval is divided into small subintervals, each contributing to the sum as a rectangle's area base times height.
• The sum approximates the curve’s area, with accuracy increasing as more subintervals (higher \(n\)) are used.
• The limit of the Riemann sum as \(n \rightarrow \infty\) provides the exact integral value.

Understanding this concept is vital, as it underlies the numerical and conceptual approach of integrals in calculus, forming a bridge between sums and continuous functions.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concepts of differentiation and integration, showing that they are inverse processes.
For the exercise, the theorem directly allows the evaluation of \( \int_0^x \cos(t) \, dt \) as \( \sin(x) \).
The theorem states:

• The definite integral of a function over an interval can be computed using its antiderivative.
• If \(F\) is an antiderivative of \(f\), then \( \int_a^b f(t) \, dt = F(b) - F(a) \).

This was applied as follows: since the derivative \( \frac{d}{dt} \sin(t) = \cos(t) \), the integral becomes \( \sin(x) - \sin(0) \), which simplifies to \( \sin(x) \).
This theorem is crucial, as it greatly simplifies the computation of definite integrals, converting them into problems of finding antiderivatives instead of evaluating limits of sums.