Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Limit Representation

The expression \( \lim_{\|P\| \rightarrow 0}\sum_{i=1}^{n}\left(\sin \bar{x}_i\right)^2 \Delta x_i \) is a Riemann sum. It approximates the area under the curve \( (\sin x)^2 \) over the interval \([a, b]\). Here, \(\bar{x}_i\) is a sample point within the subinterval \([x_{i-1}, x_i]\) and \(\Delta x_i\) is the width of the subinterval.

Step 2: Identify the Function and Interval

Recognize that the function being considered is \(f(x) = (\sin x)^2\). The interval is given by \([a, b] = [0, \pi]\) with \(a = 0\) and \(b = \pi\).

Step 3: Translate the Limit to Definite Integral

The limit expression of a Riemann sum becomes a definite integral as \(\|P\|\) approaches 0. Therefore, the given limit expression translates to \(\int_{0}^{\pi} (\sin x)^2 \, dx\). This is the area under the curve \((\sin x)^2\) from \(x = 0\) to \(x = \pi\).

Key Concepts

Riemann Sum

When learning about the definite integral, the Riemann sum is a fundamental concept. It helps in approximating the area under a curve by breaking down the area into small rectangles. Consider a function \(f(x)\) defined on an interval \([a, b]\). Divide this interval into \(n\) equally spaced subintervals. Each rectangle has a width \(\Delta x = \frac{b-a}{n}\). The height of each rectangle is given by the function value at a chosen point within the subinterval.
Riemann sums take the form:

• \( \sum_{i=1}^{n} f(\bar{x}_i) \Delta x \)

Here, \(\bar{x}_i\) is a point in the \(i\)-th subinterval, which can be the left endpoint, right endpoint, or midpoint. As the number of subintervals increases, our approximation becomes more accurate, which naturally leads us to consider the limit as \(n\) approaches infinity.

Limit Expression

The limit expression in the context of integral calculus is crucial. It shows how Riemann sums convert into definite integrals. When we take the limit of a Riemann sum as the norm of the partition \(\|P\|\) approaches zero, we essentially mean that the width of these rectangles becomes infinitely small. This allows the sum of their areas to accurately represent the entire area under the curve. In our exercise, the limit expression is:

• \( \lim _{\|P\| \rightarrow 0} \sum_{i=1}^{n} (\sin \bar{x}_i)^2 \Delta x_i \)

As \(\|P\|\) gets closer to zero, this transforms into the integral:

• \( \int_{0}^{\pi} (\sin x)^2 \, dx \)

This transformation shows the power of limits in calculus, enabling us to evaluate continuous phenomena.

Trigonometric Functions

Trigonometric functions like \(\sin(x)\) and \(\cos(x)\) play a major role in various mathematical applications, including calculus. In our context, we are dealing with \(\sin^2(x)\), a function which is commonly encountered in problems involving periodic phenomena and waveforms.
These functions have unique properties such as periodicity and symmetry, which make them interesting for analysis. The function \(\sin^2(x)\) can be rewritten using a trigonometric identity:

• \( \sin^2(x) = \frac{1}{2}(1 - \cos(2x)) \)

This identity often simplifies integration or calculation while dealing with such expressions. Understanding trigonometric functions and their identities can be extremely helpful when working on integrals involving these functions.

Area Under a Curve

The area under a curve is a significant concept in calculus. It helps quantify the accumulation of quantities, such as distance, area, or volume. Calculating the area under a curve is a primary application of definite integrals.
In the given exercise, the aim is to find the area under the curve \((\sin x)^2\) from 0 to \(\pi\). This involves transforming a Riemann sum into a definite integral. Specifically, the problem leads us to solve:

• \( \int_{0}^{\pi} (\sin x)^2 \, dx \)

By evaluating this integral, one finds the exact area under the curve, which represents the accumulation of the squares of the sine values over the specified interval. It's a powerful method to solve real-world problems ranging from physics to economics, involving accumulation over time or space.