Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 7-10, use the given values of a and b and express the given limit as a definite integral. \(\lim _{\| P \mid \rightarrow 0} \sum_{i=1}^{n}\left(\sin \bar{x}_{i}\right)^{2} \Delta x_{i} ; a=0, b=\pi\)

Step-by-Step Solution

Answer

The limit is expressed as \( \int_{0}^{\pi} (\sin x)^2 \, dx \).

Step 1: Identify the Mathematical Representation

The limit expression provided is an approximation to a definite integral of a function over the interval from \(a\) to \(b\). In this case, the function is \((\sin \bar{x}_{i})^2\) and the interval is from \(0\) to \(\pi\).

Step 2: Recognize the Riemann Sum Structure

A Riemann sum is generally written as \( \lim _{\| P \| \rightarrow 0} \sum_{i=1}^{n} f(\bar{x}_{i}) \Delta x_{i} \). Here, the function \(f(x)\) is \((\sin x)^2\), and the interval from \(a\) to \(b\) is \(0\) to \(\pi\).

Step 3: Express as a Definite Integral

The Riemann sum \( \lim _{\| P \| \rightarrow 0} \sum_{i=1}^{n} (\sin \bar{x}_{i})^2 \Delta x_{i} \) converts to the definite integral \( \int_{0}^{\pi} (\sin x)^2 \, dx \).

Step 4: Write the Final Answer

The given limit can be expressed as the definite integral \( \int_{0}^{\pi} (\sin x)^2 \, dx \).

Key Concepts

Riemann sum

A Riemann sum helps approximate the value of a definite integral. Think of it like chopping a curve into many tiny rectangles. Each rectangle’s height corresponds to the value of a function at a specific point. The width is a small slice of the horizontal axis (often called the partition). When you add up the areas of these rectangles, you get an estimation of the area under the curve.

• The Riemann sum has two main components: function value and partition width.
• As the partitions get infinitely narrow (\(\| P \| \rightarrow 0\)), the Riemann sum better approximates the integral.
• The process of increasing the number of partitions while making them infinitely thin leads to a limit that defines the definite integral.

In this exercise, the limit given is a Riemann sum for the function \((\sin \bar{x}_{i})^2\), over the interval from \(0\) to \(\pi\). Each part of the sum, \((\sin \bar{x}_{i})^2 \Delta x_{i}\), represents the area of a small rectangle under the curve.

trigonometric functions

Trigonometric functions are fundamental in math, especially in dealing with periodic phenomena like waves. Among these, the sine function (\(\sin x\)) is particularly interesting. It begins at zero, peaks at one, returns to zero, dips to negative one, and back again as \(x\) progresses along the horizontal axis. This wave-like behavior happens in a repetitive pattern, known as its period.

• The sine function is continuous and smooth, making it nice to work with in calculus.
• Squaring the sine function, as seen in \((\sin \bar{x}_{i})^2\), results in values between zero and one, smoothing the typical oscillating wave of \(\sin x\).
• Understanding how trigonometric functions behave is key to solving integrals involving them.

In the context of this exercise, dealing with \((\sin x)^2\) implies working with a function that is always non-negative over its period, which makes calculating the area below the curve straightforward.

integral calculus

Integral calculus is the study of summing things up, often areas or volumes. It's a way to handle problems involving accumulation. The definite integral, in particular, represents the area under a curve from one point to another on the horizontal axis.

• The notation \( \int_{a}^{b} f(x) \, dx \) describes an integral calculated from \(x = a\) to \(x = b\).
• The definite integral takes into account both the function and the limit of integration over the specified interval.
• When a Riemann sum reaches its limit, it transforms into a definite integral, giving the exact area under a curve.

In this problem, the integral \( \int_{0}^{\pi} (\sin x)^2 \, dx \) calculates the precise area under the curve \((\sin x)^2\) between \(0\) and \(\pi\). This represents the complete transition from an approximation (Riemann sum) to an exact mathematical description (definite integral).