Problem 46 - Calculus for Biology and Medicine

Question

As definite integrals. Note that (1) \(P=\left[x_{0}, x_{1}, \ldots, x_{n}\right]\) is a partition of the indicated interval, (2) \(c_{k} \in\left[x_{k-1}, x_{k}\right]\), and (3) \(\Delta x_{k}=x_{k}-x_{k-1}\). \(\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(\sin c_{k}\right) \Delta x_{k}\), where \(P\) is a partition of \([0, \pi]\)

Step-by-Step Solution

Answer

The value of the definite integral is 2.

Step 1: Understand the limit of sums definition

The limit expression given is a form of a Riemann sum, which can be used to find the definite integral of a function. Specifically, as the norm \(||P||\) of the partition approaches 0, the sum \(\sum_{k=1}^{n}\left(\sin c_{k}\right) \Delta x_{k}\) approaches the definite integral of \(\sin(x)\) over the interval \([0, \pi]\).

Step 2: Set up the definite integral

Identify the function being integrated, which is \(\sin(x)\), and the interval of integration, which is \([0, \pi]\). This sets up the definite integral \(\int_{0}^{\pi} \sin(x) \, dx\).

Step 3: Integrate the function

The integral of \(\sin(x)\) with respect to \(x\) is \(-\cos(x)\). Therefore, evaluate the integral: \[ \int_{0}^{\pi} \sin(x) \, dx = [-\cos(x)]_{0}^{\pi}. \]

Step 4: Evaluate the integral at the bounds

Substitute the upper bound \(\pi\) and lower bound \(0\) into the antiderivative:\[ -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 1 + 1 = 2. \]

Step 5: Conclude the result

The value of the definite integral is 2, which means that the original limit of Riemann sums evaluates to 2 as \(||P||\) approaches zero.

Key Concepts

Understanding Riemann Sums

Riemann sums are a fundamental concept in calculus used to approximate the area under a curve, which is essentially what an integral does. Imagine slicing the area under a curve into many thin rectangles. By summing up the areas of these rectangles and taking the limit as their width approaches zero, we calculate the exact area. This is why you might often hear that Riemann sums "approach the integral." The Riemann sum formula can be expressed as \( \sum_{k=1}^{n} f(c_k) \Delta x_k \), where \( f(c_k) \) is the function value at a point \( c_k \) within each subinterval of the partition, and \( \Delta x_k \) is the width of the interval.

• The partition \( P = [x_0, x_1, \ldots, x_n] \) divides the interval into smaller parts.
• The width of the largest subinterval is represented by \( ||P|| \).
• Riemann sums become more accurate as \( ||P|| \) becomes smaller, effectively turning the sum into a definite integral.

By transforming the Riemann sum into an integral, we bridge the discrete and the continuous, transitioning from a sum of individual parts to a smooth, continuous area calculation.

Exploring Integration Techniques

Integration is the key process in calculus for finding the exact area under a curve. Different functions require different techniques. When integrating \( \sin(x) \), a standard technique involves finding an antiderivative of the function. Since the derivative of \(-\cos(x)\) is \( \sin(x) \), \(-\cos(x)\) is the antiderivative we use.After setting up the integral \( \int_{0}^{\pi} \sin(x) \, dx \), we:

• Find the antiderivative: \( -\cos(x) \).
• Apply the Fundamental Theorem of Calculus, evaluating \( -\cos(x) \) at the bounds \( 0 \) and \( \pi \).
• Subtract: \( -\cos(\pi) - (-\cos(0)) = 1 + 1 = 2 \).

This specific integration process demonstrates how the setup and solution approach depends significantly on the function itself, encouraging understanding of various techniques to integrate different forms. The application of these techniques allows us to solve complex problems involving areas and accumulated quantities.

Investigating the Sine Function

The sine function, \( \sin(x) \), is one of the basic trigonometric functions, representing periodic oscillations. It is used extensively in modeling wave patterns and cycles. The graph of \( \sin(x) \) forms a smooth wave-like curve oscillating between -1 and 1 as \( x \) increases. When examining the sine function over an interval like \([0, \pi]\), we observe half of its oscillation cycle.

• The maximum value of \( \sin(x) \) is 1, occurring at \( \frac{\pi}{2} \), the midpoint of \([0, \pi]\).
• The minimum value is 0 at the ends \(0\) and \(\pi\).

This function is particularly interesting in calculus since it's periodic and smooth, making it a suitable candidate for illustrating integration and differentiation techniques. Understanding the properties of \(\sin(x)\) helps in visualizing how its integration relates to the area under its curve, directly connected to the geometry of its wave form.