Problem 13 - Calculus Volume 1

Question

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$R_{4} \text { for } g(x)=\cos (\pi x) \text { on }[0,1]$$

Step-by-Step Solution

Answer

The right-endpoint sum \(R_4\) is \(-\frac{1}{4}\).

Step 1: Identify the Interval and Number of Subdivisions

We are asked to compute the right-endpoint sum, \(R_4\), for the function \(g(x) = \cos(\pi x)\) on the interval \([0, 1]\). This means we divide the interval into \(n=4\) equal parts.

Step 2: Calculate the Width of Each Subinterval

The width of each subinterval is given by \( \Delta x = \frac{b-a}{n} \), where \(a = 0\), \(b = 1\), and \(n = 4\). Therefore, \(\Delta x = \frac{1-0}{4} = \frac{1}{4}\).

Step 3: Determine the Right-endpoints of the Subintervals

With the interval \([0, 1]\) divided into 4 parts, the right endpoints will be \(x_1 = \frac{1}{4}, x_2 = \frac{2}{4}, x_3 = \frac{3}{4}, x_4 = 1\).

Step 4: Evaluate the Function at Each Right-endpoint

Substitute each right endpoint into \(g(x) = \cos(\pi x)\):- \(g\left(\frac{1}{4}\right) = \cos\left(\frac{\pi}{4}\right)\)- \(g\left(\frac{1}{2}\right) = \cos\left(\frac{\pi}{2}\right)\)- \(g\left(\frac{3}{4}\right) = \cos\left(\frac{3\pi}{4}\right)\)- \(g(1) = \cos(\pi)\).The evaluations are:- \(g\left(\frac{1}{4}\right) = \frac{\sqrt{2}}{2}\)- \(g\left(\frac{1}{2}\right) = 0\)- \(g\left(\frac{3}{4}\right) = -\frac{\sqrt{2}}{2}\)- \(g(1) = -1\).

Step 5: Compute the Right-endpoint Sum

The right-endpoint sum \(R_4\) is given by\[R_4 = \Delta x \cdot \left[ g\left(\frac{1}{4}\right) + g\left(\frac{1}{2}\right) + g\left(\frac{3}{4}\right) + g(1) \right]\]Substituting the values:\[R_4 = \frac{1}{4} \left( \frac{\sqrt{2}}{2} + 0 - \frac{\sqrt{2}}{2} - 1 \right)\]This simplifies to:\[R_4 = \frac{1}{4} \left( 0 - 1 \right) = -\frac{1}{4}\]

Step 6: Final Result

The computed right-endpoint sum \(R_4\) for the function \(g(x) = \cos(\pi x)\) on \([0,1]\) is \(-\frac{1}{4}\).

Key Concepts

Right-Endpoint Sum

When we talk about Riemann sums, there are different ways to approach the intervals within a function. One method is using the right-endpoint sum. This involves evaluating the function at the right end of each subinterval. By using the right endpoints, we ensure each rectangle drawn over the subinterval represents an upper or lower sum, depending on the function's behavior. The formula for the right-endpoint sum, generally, is given by:

• \[R_n = \Delta x \cdot \sum_{i=1}^{n} f(x_i)\]

In this formula, \(\Delta x\) is the width of each subinterval, and \(x_i\) is the right-endpoint of the \(i\)th subinterval. Computing the right-endpoint sum helps approximate the area under a curve when it's difficult to calculate the exact integral.

Subintervals in Integration

Subintervals in integration are integral for dissecting the interval over which we want to find the area under a curve. By dividing the overall interval into smaller subintervals, we simplify our calculations.

• In general, the width of each subinterval \(\Delta x\) is calculated as:\[\Delta x = \frac{b-a}{n}\]Here, \(a\) and \(b\) are the interval's endpoints, and \(n\) is the number of subintervals.
• Using the right endpoints, as in the case above with four subintervals on \([0,1]\), we determine the values at \(x_1 = \frac{1}{4}, x_2 = \frac{2}{4}\), and so on.

This step-by-step division means seeing the effect of a function in small, manageable chunks, therefore allowing an approximate total area that closely mirrors calculus-derived solutions.

Trigonometric Functions

Trigonometric functions, such as cosine and sine, are foundational in calculus, especially when working with periodic phenomena or circular motion. Here, our function \(g(x) = \cos(\pi x)\) embodies such a trigonometric function.

• These functions have characteristic waves and patterns:
• The cosine function starts at its maximum value, decreases to zero, then dives into the negatives, forming a perfect oscillation.
• For example, the value \(\cos(\pi)\) equals -1, showing periodic and oscillatory behavior.

Understanding the cosine function is vital for predicting such behavior and calculating accurate sums in problems involving rotation and wave motion. That's why evaluating each endpoint's function value is crucial—ensuring these natural oscillations and periodic changes are accounted for in a Riemann sum.